Abstract. A well-known result, due to Meyer, states that the set \(\wp _{K}\)of Pisot numbers, generating a real algebraic number field \(K,\)is uniformly discrete and relatively dense in the interval \([1,\infty ).\)In the present paper, we show that \(\wp _{K}\cup \{1\}\subset \wp _{K}-\wp _{K}\)and the complement of \(\wp _{K}\) in \(\wp _{K}-\wp _{K}\)is not finite. Also, we provethat if \(K\)is totally real, then the elements of \(\wp _{K}-\wp _{K}\) are the algebraic integers of \(K\)whose images under the action of all embeddings of \(K\)into \(\mathbb{R},\)other than the identity of \(K,\)belong to the interval \((-2,2),\) and so any Salem trace number \(\beta\) may be written as a difference of two Pisot numbers generating the field \(\mathbb{Q}(\beta ).\)

1. Introduction

A Pisot number is a real algebraic integer greater than \(1\) whose other conjugates are of modulus less than \(1,\) and the set of such numbers is traditionally denoted by \(S.\) Let throughout \(K\) denote a real algebraic number field of degree \(d,\) and let \[\wp _{K}:=\{\theta \mid \theta \in S\cap K,\text{ \;}K=\mathbb{Q}(\theta )\}.\]Since there are at most a finite number of algebraic integers of fixed degree and having all conjugates in a bounded subset of the complex plane, the set of Pisot numbers of a given degree is a discrete subset of \([1,\infty ),\) i. e., any finite subinterval of \([1,\infty )\) contains at most a finite number of such numbers, and so the family \(\wp _{K}\) is also discrete in \([1,\infty ).\)

Pisot [13] was the first to investigate the set \(\wp _{K}\) and he showed, in particular, that \(\wp _{K}\) contains units, whenever \(d\geq 2.\) Fan and Schmeling [7] have proved that \(\wp _{K}\) is relatively dense in \([1,\infty ),\) that is, there is a real number \(\rho >0\) such that every subinterval of \([1,\infty )\) of the form \((\varepsilon ,\varepsilon +\rho ]\) contains at least one element of \(\wp _{K};\) to be more precise, we say in this case that \(\wp _{K}\) is \(\rho\)-dense in \([1,\infty ).\)

Using some theorems, due to Meyer, on harmonious sets (see for instance [8], [10], and [12]), the author pointed out, in [19], that the family \(\wp _{K}^{\prime }:=\wp _{K}\cup (-\wp _{K})\) is a real Meyer set, i. e., there is a finite subset \(F\) of the real line \(\mathbb{R}\) such that \(\wp _{K}^{\prime }-\wp _{K}^{\prime }\subset \wp _{K}^{\prime }+F,\) and a generalization of this result for complex Pisot numbers was given in [2]. A real Meyer set is, in particular, a relatively dense subset of \(\mathbb{R}\), which is also uniformly discrete, that is, there is a positive real number \(\rho\) such that each real interval of the form \((\varepsilon ,\varepsilon +\rho ]\) contains at most one element of this set [12].

It is worth noting that the set of Pisot numbers of a given degree \(d\geq 2,\)is not uniformly discrete. Indeed, it easy to see, by Rouche’s theorem, that a polynomial of the form \(x^{d}-nx^{d-1}-k,\) where \(k\in \{1,2\}\) and \(n\in \lbrack 4,\infty )\cap \mathbb{N},\) is the minimal polynomial of a Pisot number lying in \((n,n+k/n),\) and so the interval \((n,n+2/n)\) contains two distinct Pisot numbers of degree \(d.\)

Let \(\theta _{1}<\theta _{2}<\theta _{3}<\cdot \cdot \cdot\) denote the elements of \(\wp _{K},\) \[D_{K}:=\{\theta _{m}-\theta _{n}\mid (m,n)\in \mathbb{N}^{2},\text{ }n>m\},\]and

\[\tciFourier _{K}:=\{\theta _{n+1}-\theta _{n}\mid \text{ }n\in \mathbb{N}\}.\]Then, \(\wp _{K}-\wp _{K}=(-D_{K})\cup \{0\}\cup D_{K}\) and \[\tciFourier _{K}\subset D_{K}\subset E_{K}:=\{\beta \in \mathbb{Z}_{K}\cap (0,\infty )\mid (\left\vert \sigma _{2}(\beta )\right\vert ,...,\left\vert \sigma _{d}(\beta )\right\vert )\in (0,2)^{d-1}\}, \]where \(\mathbb{Z}_{K}\) is the ring of the integers of \(K,\) and \(\sigma _{2},...,\sigma _{d}\) are the distinct embeddings of \(K\mathbb{\;}\)into \(\mathbb{C}\) other than the identity of \(K.\)

Because the set \(\wp _{K}\) is \(\rho\)-dense in \([1,\infty ),\) for some \(\rho >0,\) we have \((\theta _{n},\theta _{n}+\rho ]\cap \wp _{K}\neq \varnothing\) for each \(n,\) and so \(\theta _{n+1}-\theta _{n}\leq \rho .\) Further, as \(\tciFourier _{K}\subset E_{K},\) the degree and the absolute values of the conjugates of each element of \(\tciFourier _{K}\) are, respectively, majored by \(d\) and \(\max (\rho ,2),\) and hence the set \(\tciFourier _{K}\) is finite.

Let\[d_{1}<\cdot \cdot \cdot <d_{\func{card}(\tciFourier _{K})}\]be the elements of \(\tciFourier _{K}.\) Then, \(\wp _{K}\) is \(\max \{\theta _{1}-1,\) \(d_{\func{card}(\tciFourier _{K})}\}\)-dense in \([1,\infty ),\) and \(\max \{\theta _{1}-1,\) \(d_{\func{card}(\tciFourier _{K})}\}\) is the infimum of those numbers \(\rho\) for which \(\wp _{K}\) is \(\rho\)-dense in \([1,\infty ).\) Also, the distance between any two distinct elements of \(\wp _{K}\) is at least \(d_{1}\) (this implies that \(\wp _{K}\) is uniformly discrete), and each element of \(D_{K}\) may be written \[\theta _{n}-\theta _{m}=(\theta _{n}-\theta _{n-1})+(\theta _{n-1}-\theta _{n-2})+\cdot \cdot \cdot +(\theta _{m+1}-\theta _{m})=l_{1}d_{1}+\cdot \cdot \cdot +l_{\func{card}(\tciFourier _{K})}d_{\func{card}(\tciFourier _{K})}, \]for some non-negative integers \(l_{1},\) \(...,\) \(l_{\func{card}(\tciFourier _{K})},\) not all zero.

For example, if \(K=\mathbb{Q},\) then \(\wp _{\mathbb{Q}}=\mathbb{N\diagdown \{}1\},\) \(\wp _{\mathbb{Q}}-\wp _{\mathbb{Q}}=\mathbb{Z},\) \(D_{\mathbb{Q}}=\mathbb{N}=E_{\mathbb{Q}}\) and \(\tciFourier _{\mathbb{Q}}=\{1\}.\) It is of interest to determine the elements of the sets \(\tciFourier _{K}\) and \(D_{K}.\)

Theorem 1.1 With the notation above, we have the following.

(i) \(1\in D_{K};\)

(ii) \(\wp _{K}\varsubsetneq D_{K};\)

(iii) if \(K\neq \mathbb{Q},\)then the complement of \(\wp _{K}\)in \(D_{K}\)is not finite, and \(\func{card}(\tciFourier _{K})\geq 2.\)

Theorem 1.1(i) implies the result below, shown recently by Dubickas in [5].

Corollary 1.2 [5] If \(K=\mathbb{Q}(\tau )\)for some Salem number \(\tau ,\)then \(\tau \in D_{K}.\)

Recall that a Salem number is a real algebraic integer \(\tau >1\) whose other conjugates are of modulus at most \(1,\) with at least one conjugate of modulus \(1.\) Then, \(\tau\) has two real conjugates, namely \(\tau\) and \(\tau ^{-1},\) and \(\deg (\tau )-2\geq 2\) conjugates, say \(\tau _{2}^{\pm 1},...,\tau _{d}^{\pm 1},\) lying on the unit circle. Also, the algebraic integer \(\tau +\tau ^{-1}>2,\) called a Salem trace number [20], is of degree \(d,\) and its other conjugates are the numbers \(\tau _{2}+\tau _{2}^{-1},\) \(...,\) \(\tau _{d}+\tau _{d}^{-1},\) lying in the interval \((-2,2).\)

Conversely, an algebraic integer \(\beta >2\) of degree \(d\geq 2,\) whose other conjugates belong the interval \((-2,2),\) is a Salem trace number associated to some Salem number \(\tau\) of degree \(2d,\) via the relation \(\beta =\tau +\tau ^{-1}\) (for more details, see the proofs of Lemmas 3.1 and 3.3).

By analogy with Salem trace numbers, we call a positive algebraic integer of the form \(u:=e^{i2k\pi /n}+e^{-i2k\pi /n}=2\cos (2k\pi /n),\) where \(n\in \mathbb{N}\cap \lbrack 4,\infty ),\) \(k\in \{1,...,n-1\}\) and \(\gcd (k,n)=1,\) a root of unity trace number. Then, \(\deg (u)=\varphi (n)/2,\) where \(\varphi\) is the Euler totient function, and the conjugates of \(u\) are the numbers \(2\cos (2l\pi /n)\in (-2,2),\) where \(l\in\) \(\{1,...,[n/2]\},\) \([.]\) is the integer part function, and \(\gcd (l,n)=1.\)

Clearly, the set, say \(U_{K},\) of root of unity trace numbers lying in \(K\) is finite, and \(1=2\cos (2\pi /6)\in U_{K};\) we denote by \[u_{1}<\cdot \cdot \cdot <u_{\func{card}(U_{K})}\]the elements of \(U_{K}.\)

For the case where the field \(K\) is totally real, i.e., the conjugates of \(K\) are all real, the set of Salem trace numbers generating \(K\) over \(\mathbb{Q},\) and the set \(U_{K}\) form a partition of \(D_{K},\) as stated by the following result.

Theorem 1.3 Let \(K\)be a totally real number field of degree \(d\geq 2.\)Then, the assertions below are true.

(i) \(D_{K}=E_{K};\)

(ii) \(U_{K}\)\(=E_{K}\cap (0,2)=D_{K}\cap (0,2);\)

(iii) the sets \(T_{K}:=E_{K}\cap (2,\infty )=D_{K}\cap (2,\infty )\)and \(U_{K}\) form a partition of \(D_{K},\) and the degree of each element of \(T_{K}\)is equal to \(d;\)

(iv) \(\func{card}(\tciFourier _{K})\geq 2^{d-1};\)

(v) \(\tciFourier _{K}\cap (0,2)\subset U_{K},\) \(d_{1}=u_{1}\leq 1,\) and \(d_{2}=u_{2}\) whenever \(\func{card}(U_{K})\geq 2;\)

(vi) if \(\func{card}(U_{K})=1,\) then \(d_{1}=1\) and \(d_{2}=\min T_{K}\)is the smallest Salem trace number of degree \(d,\)lying in \(K.\)

Notice that Theorem 1.3(vi) holds, for example, when \(d\) is prime and the discriminant of the field \(K\) is sufficiently large or when \(d\) is prime and \(2d\) is not a totient number, so that the field \(K\) cannot be embedded in a cyclotomic field.

In the next section, we determine explicitly the elements of \(\wp _{K},\) \(\tciFourier _{K}\) and \(U_{K},\) when \(K\) is a real quadratic field. The proofs of Theorem 1.1 and its corollary are given in Section 4, and the proof of Theorem 1.3 is postponed to the last section. These proofs are based on some lemmas presented in Section 3.

Throughout, when we speak about conjugates, the norm and the degree of an algebraic number, without mentioning the basic field, this is meant over \(\mathbb{Q}.\) Also, the degree, the discriminant and the conjugates of a number field are considered over \(\mathbb{Q},\) and an integer means a rational integer. All computations are done using the system Pari [15].

2. The quadratic case

Let \(K=\mathbb{Q}(\sqrt{m})\) be a real quadratic field, where \(m\equiv 2,3\) \(\func{mod}4\) is a square-free integer, and let \(\theta \in \wp _{K}.\) Then, \(\theta =a+b\sqrt{m}\) for some \((a,b)\in \mathbb{Z}\times \mathbb{Z\diagdown }\{0\},\) and the inequalities \(a+b\sqrt{m}>1\) and \(\left\vert a-b\sqrt{m}\right\vert <1\) yield \(b\geq 1,\) \(a\in \{\left\lfloor b\sqrt{m}\right\rfloor ,\left\lfloor b\sqrt{m}\right\rfloor +1\},\) and so \[\theta =\alpha _{b}:=\left\lfloor b\sqrt{m}\right\rfloor +b\sqrt{m}\text{ \; \;or \;\;}\theta =\widehat{\alpha }_{b}:=1+\alpha _{b}\}.\]It is clear that for each \(b\in \mathbb{N},\) \(\{\alpha _{b},\widehat{\alpha }_{b}\}\subset \wp _{K},\) \(\widehat{\alpha }_{b}-\alpha _{b}=1,\) and \(\alpha _{b+1}-\widehat{\alpha }_{b}=\left\lfloor (b+1)\sqrt{m}\right\rfloor +(b+1)\sqrt{m}-(1+\left\lfloor b\sqrt{m}\right\rfloor +b\sqrt{m})=\left\lfloor (b+1)\sqrt{m}\right\rfloor +\sqrt{m}-(1+\left\lfloor b\sqrt{m}\right\rfloor ).\) Because \(\left\lfloor b\sqrt{m}\right\rfloor +\left\lfloor \sqrt{m}\right\rfloor <(b+1)\sqrt{m}<\left\lfloor b\sqrt{m}\right\rfloor +\left\lfloor \sqrt{m}\right\rfloor +2,\) we see that \(\left\lfloor (b+1)\sqrt{m}\right\rfloor -\left\lfloor b\sqrt{m}\right\rfloor \in \{\left\lfloor \sqrt{m}\right\rfloor ,\left\lfloor \sqrt{m}\right\rfloor +1\}\) and \[\alpha _{b+1}-\widehat{\alpha }_{b}\in \{\left\lfloor \sqrt{m}\right\rfloor -1+\sqrt{m},\text{ }\left\lfloor \sqrt{m}\right\rfloor +\sqrt{m}\}.\]Therefore, the elements of \(\wp _{K}\) may be labelled as follows:\[\theta _{1}:=\alpha _{1}<\theta _{2}:=\widehat{\alpha }_{1}<\theta _{3}:=\alpha _{2}<\theta _{4}:=\widehat{\alpha }_{2}<\theta _{5}:=\alpha _{3}<\cdot \cdot \cdot\]Since the sequence \((b\sqrt{m})_{b\in \mathbb{N}}\) is dense modulo one (this also follows from Lemma 3.1 with \(n=1\)), we get from the equivalences \(\alpha _{b+1}-\widehat{\alpha }_{b}=\left\lfloor \sqrt{m}\right\rfloor -1+\sqrt{m}\Leftrightarrow b\sqrt{m}-\left\lfloor b\sqrt{m}\right\rfloor <1+\) \(\left\lfloor \sqrt{m}\right\rfloor -\sqrt{m}\) and \(\;\alpha _{b+1}-\widehat{\alpha }_{b}=\left\lfloor \sqrt{m}\right\rfloor +\sqrt{m}\Leftrightarrow b\sqrt{m}-\left\lfloor b\sqrt{m}\right\rfloor >1+\left\lfloor \sqrt{m}\right\rfloor -\sqrt{m}\) that \[\{\alpha _{b+1}-\widehat{\alpha }_{b}\mid b\in \mathbb{N}\}=\{\left\lfloor \sqrt{m}\right\rfloor -1+\sqrt{m},\text{ }\left\lfloor \sqrt{m}\right\rfloor +\sqrt{m}\},\]and so \[\tciFourier _{K}=\{1,\text{ }\left\lfloor \sqrt{m}\right\rfloor -1+\sqrt{m},\text{ }\left\lfloor \sqrt{m}\right\rfloor +\sqrt{m}\}.\]Finally, notice that if \(a+b\sqrt{m}\in\) \(U_{K}\) for some \((a,b)\in \mathbb{Z}^{2},\) then \(a+b\sqrt{m}\in (0,2),\) \(a-b\sqrt{m}\in (-2,2),\) and so \((a,b)=(1,0)\) (resp. \((a,b)\in \{(1,0),(0,1)\})\) when \(m\geq 6\) (resp. when \(m\in \{2,3\}).\) Therefore, we have the following.

Proposition 2.1 Let \(K=\mathbb{Q}(\sqrt{m})\)be a real quadratic field, where \(m\equiv 2,3\)\(\func{mod}4\) is a square-free integer. If \(m=2\) (resp. \(\;m=3,\) \(m\geq 6),\) then\[U_{K}=\{1,\sqrt{2}=2\cos (2\pi /8)\}\subset \tciFourier _{K}=\{1,\sqrt{2},\text{ }1+\sqrt{2}=\min T_{K}\}\](resp. \[U_{K}=\{1,\sqrt{3}=2\cos (2\pi /12)\}\subset \tciFourier _{K}=\{1,\sqrt{3},1+\sqrt{3}=\min T_{K}\},\]\[U_{K}=\{1\}\subset \tciFourier _{K}=\{1,\text{ }\left\lfloor \sqrt{m}\right\rfloor -1+\sqrt{m}=\min T_{K},\text{ }1+\min T_{K}\}).\]In fact, the equality \(\min T_{K}=\left\lfloor \sqrt{m}\right\rfloor -1+\sqrt{m}\)(resp. \(\min T_{K}=1+\sqrt{2},\) \(\min T_{K}=1+\sqrt{3}),\) for \(m\geq 6\) (resp. for \(\;m=2,\) for \(m=3)\) follows from Theorem 1.3(vi) (resp. from a simple computation).

In the same way, we obtain the following assertion.

Proposition 2.2 Let \(K=\mathbb{Q}(\sqrt{m})\)be a real quadratic field, where \(m\equiv 1\)\(\func{mod}4\)is a square-free integer. Then, \[U_{K}=\{1\}\subset \tciFourier _{K}=\{1,\text{ }\frac{-3+\left\lfloor \sqrt{m}\right\rfloor +\sqrt{m}}{2}=\min T_{K},\text{ }1+\min T_{K}\}\]when \(\left\lfloor \sqrt{m}\right\rfloor\)is an even integer greater than \(2,\) \[U_{\mathbb{Q}(\sqrt{5})}=\tciFourier _{\mathbb{Q}(\sqrt{5})}=\{\frac{-1+\sqrt{5}}{2}=2\cos (\frac{2\pi }{5}),\text{ }1,\text{ }\frac{1+\sqrt{5}}{2}=2\cos (\frac{2\pi }{10})\},\]and \[U_{K}=\{1\}\subset \tciFourier _{K}=\{1,\text{ }\frac{-2+\left\lfloor \sqrt{m}\right\rfloor +\sqrt{m}}{2}=\min T_{K},1+\min T_{K}\}\]when \(\left\lfloor \sqrt{m}\right\rfloor\)is odd.

Finally, notice that a short calculation gives that \(\min T_{\mathbb{Q}(\sqrt{5})}=(3+\sqrt{5})/2,\) and it follows, by Propositions 2.1 and 2.2, that \[\min_{K\text{ real quadratic field }}(\min T_{K})=\frac{1+\sqrt{13}}{2};\]thus the smallest quartic Salem number \(\tau _{0}\) satisfies the (well-known) equation \(\tau _{0}+1/\tau _{0}=(1+\sqrt{13})/2.\)

Remark 2.3 Let \(K\) be a totally real cubic field and let \(\alpha \in U_{K}.\) Then, \(\alpha =1\) or \(\mathbb{Q}(\alpha )=K.\) Suppose \(\alpha\) cubic. Then, \(\alpha\) is a conjugate of \(2\cos (\frac{2\pi }{n}),\) where \(\varphi (n)=6.\) Hence, \(n\in \{7,14,9,18\},\) and if \(n\in \{7,14\}\) (resp. \(n\in \{9,18\}),\) then the discriminant \(\func{disc}(K)\) of \(K\) is equal to \(49\) and \[U_{K}=\{-2\cos (\frac{4\pi }{7}),1,2\cos (\frac{2\pi }{7}),-2\cos (\frac{6\pi }{7})\}\] (resp. is equal to \(81\) and \[U_{K}=\{2\cos (\frac{4\pi }{9}),1,2\cos (\frac{2\pi }{9}),-2\cos (\frac{8\pi }{9})\}).\] Also, if \(\func{disc}(K)\notin \{49,81\},\) then \(U_{K}=\{1\}.\)

Since the smallest Salem number of degree \(6,\) say \(\tau ,\) is given by the equality \(\tau ^{6}-\tau ^{4}-\tau ^{3}-\tau ^{2}+1=0,\) the smallest cubic Salem trace number \(\beta =\tau +1/\tau\) satisfies \(\beta ^{3}-4\beta -1=0\) and \(\beta =2.1149...\). Also, because the discriminant of the polynomial \(x^{3}-4x-1\) is equal to the prime number \(229,\) we see that \(\func{disc}(\mathbb{Q}(\beta ))=229,\) \(U_{\mathbb{Q}(\beta )}=\{1\},\) and so, by Theorem 1.3(vi), the two smallest element of \(\tciFourier _{\mathbb{Q}(\beta )}\) are \(1\) and \(\beta .\)

3. Some Lemmas

In the form in which we use it here, where all numbers are real, Kronecker’s theorem on linearly independent numbers may be stated as follows (see also [3] and [14, page 66]).

Lemma 3.0 [8] If \(1,\)\(\omega _{2},\)\(...,\)\(\omega _{v}\)are \(\mathbb{Q}\)-linearly independent, \(\eta _{2},\)\(...,\)\(\eta _{v}\) are arbitrary, and \(\rho\)and \(\varepsilon\)are positive, then there exist integers \(p_{2},\) \(...,\)\(p_{v},\)\(n>\rho\)such that \(\max_{2\leq j\leq v}\left\vert n\omega _{j}-p_{j}-\eta _{j}\right\vert <\varepsilon .\)

Proof. See for instance [4, Chapter 4].

The first part of the following simple lemma is given in [15, Proposition 3(i)].

Lemma 3.1 Let \(\beta >2\)be a real algebraic integer of degree \(d\geq 2\)whose other conjugates lie in the interval \((-2,2).\)Then, there is a Salem number \(\tau\) of degree \(2d\)such that \(\beta =\tau +1/\tau ,\)i. e., \(\beta\) is a Salem trace number. Moreover, \(\mathbb{Q}(\beta )\) is the set of totally real algebraic numbers lying in \(\mathbb{Q}(\tau ),\)and so it is the unique totally real subfield of \(\mathbb{Q}(\tau )\)with degree \(d.\)

Proof. Let \(\beta _{1}:=\beta ,\) \(\beta _{2},...,\beta _{d}\) be the conjugates of \(\beta ,\) and let \(\tau\) be the root greater than \(1\) of the quadratic polynomial \(x^{2}-\) \(\beta x+1.\) Then, \(\tau\) is a zero of the monic polynomial \(P(x):=(x^{2}-\beta _{1}x+1)\cdot \cdot \cdot (x^{2}-\beta _{d}x+1)\in \mathbb{Z}[x],\) and so it is an algebraic integer of degree at most \(2d,\) whose conjugates are among the two real numbers \(\tau\) and \(1/\tau\) and the \(2d-2\) non-real numbers of modulus \(1\) roots of the quadratic polynomials \(x^{2}-\) \(\beta _{j}x+1,\) where \(j\in \{2,...,d\}.\) Further, as \(\tau +1/\tau =\beta ,\) we have that \(\mathbb{Q}(\beta )\subset \mathbb{Q}(\tau ),\) and so \(\deg (\tau )\in \{d,\) \(2d\}.\) In fact, if \(\deg (\tau )=d,\) then \(\mathbb{Q}(\beta )=\mathbb{Q}(\tau ),\) \(\tau\) is totally real with minimal polynomial \(x^{2}-\) \(\beta x+1,\) and this last assertion leads immediately to the contradiction \(\beta \in \mathbb{Z}.\) Therefore, \(\deg (\tau )=2d,\) \(\tau\) is a Salem number with minimal polynomial \(P,\) and \(\beta\) is a Salem trace number.

To complete the proof of the lemma, notice that any element of \(\mathbb{Q}(\beta )\) is totally real, and assume on the contrary that there is a totally real number \(\delta \in \mathbb{Q}(\tau )\) which is not in \(\mathbb{Q}(\beta ).\) Then, the relation \(\deg (\mathbb{Q}(\beta ,\delta ))=nd\leq 2d=\deg (K),\) where \(n\) is a natural number greater than \(1,\) yields \(n=2,\) \(\mathbb{Q}(\tau )\) is equal to the totally real field \(\mathbb{Q}(\beta ,\delta ),\) and this last equality contradicts the fact that \(\tau\) has non-real conjugates; thus \(\mathbb{Q}(\beta )\) contains all totally real numbers in \(\mathbb{Q}(\tau ).\) 

---

The following result is due to Pisot [13] (see also [1] and [14]) for Salem numbers, and to Boyd (unpublished) for Pisot numbers (see [1, Theorem 8.2], and [11] for some related generalizations).

Lemma 3.2 (Pisot-Boyd)Let \(e^{\pm i2\pi \omega _{2}},\) \(...,\) \(e^{\pm i2\pi \omega _{d}}\) be the non-real conjugates of a Salem number of degree \(2d.\)Then, the numbers \(1,\)\(\omega _{2},\) \(...,\) \(\omega _{d}\)are\(\mathbb{Q}\)-linearly independent.

Similarly, if \(\rho _{1}e^{\pm i2\pi \omega _{1}},\) \(...,\) \(\rho _{s}e^{\pm i2\pi \omega _{s}}\) denote the non-real conjugates of a non-totally real Pisot number, then the numbers \(1,\)\(\omega _{1},\) \(...,\) \(\omega _{s}\)are\(\mathbb{Q}\)-linearly independent.

Proof. To give a common proof for the two cases, consider a non-totally real Pisot number (resp. a Salem number) \(\alpha ,\) and let \(\rho _{2}e^{\pm i2\pi \omega _{2}},\) \(...,\) \(\rho _{v}e^{\pm i2\pi \omega _{v}}\) be the non-real conjugates of \(\alpha ,\) where \(\rho _{2}<1,\) \(...,\) \(\rho _{v}<1\) and \(v=s+1\geq 2\) (resp. where \(\rho _{2}=\cdot \cdot \cdot =\rho _{v}=1\) and \(v=d).\)

Assume on the contrary that there are integers \(l,\) \(l_{2},...,\) \(l_{v}\) not all zero such that \(l_{2}\omega _{2}+\cdot \cdot \cdot +l_{v}\omega _{v}=l.\) By replacing, if necessary, \(\omega _{j}\) by \(-\omega _{j},\) \(\forall\) \(j\in \{2,\) \(...,\) \(v\},\) we may assume that all \(l_{j}\) are non-negative. Also, by reordering, if necessary, the non-real conjugates of \(\alpha ,\) we may suppose that \(l_{2}=\max \{l_{j}\mid 2\leq j\leq v\}.\) Then, \(l_{2}\geq 1,\) \(e^{i2\pi l_{2}\omega _{2}}\cdot \cdot \cdot e^{i2\pi l_{v}\omega _{v}}=e^{i2\pi l}=1,\) \(e^{-i2\pi l_{2}\omega _{2}}\cdot \cdot \cdot e^{-i2\pi l_{v}\omega _{v}}=1,\) \(e^{i2\pi l_{2}\omega _{2}}\cdot \cdot \cdot e^{i2\pi l_{v}\omega _{v}}=e^{-i2\pi l_{2}\omega _{2}}\cdot \cdot \cdot e^{-i2\pi l_{v}\omega _{v}},\) and \[\alpha _{2}^{l_{2}}\cdot \cdot \cdot \alpha _{v}^{l_{v}}=\alpha _{v+2}^{l_{2}}\cdot \cdot \cdot \alpha _{2v}^{l_{v}}.\]where \(\alpha _{2}:=\rho _{2}e^{i2\pi \omega _{2}},\) \(...,\) \(\alpha _{v}:=\rho _{v}e^{i2\pi \omega _{v}},\) \(\alpha _{v+2}:=\rho _{2}e^{-i2\pi \omega _{2}}=\overline{\alpha _{2}},\) \(...,\) \(\alpha _{2v}:=\rho _{v}e^{-i2\pi \omega _{v}}=\overline{\alpha _{v}}.\)

Let \(\sigma\) be an automorphism of the normal closure \(\Gamma\) of \(\mathbb{Q}(\alpha )\) in \(\mathbb{C},\) sending \(\alpha _{2}\) to \(\alpha .\) Since the Galois group of \(\Gamma\) operates transitively on the conjugates of \(\alpha ,\) we have \(\left\vert \sigma (\alpha _{v+2})\right\vert <1,\) \(...,\) \(\left\vert \sigma (\alpha _{2v})\right\vert <1\) (resp. \(\sigma (\alpha _{v+2})=\sigma (\overline{\alpha _{2}})=\sigma (1/\alpha _{2})=1/\alpha <1,\) \(\left\vert \sigma (\alpha _{v+3})\right\vert =1,\) \(...,\) \(\left\vert \sigma (\alpha _{2v})\right\vert =1),\) and so \[\alpha ^{l_{2}}\dprod\limits_{j=3}^{v}\left\vert \sigma (\alpha _{j})\right\vert ^{l_{j}}=\left\vert \sigma (\alpha _{v+2})\right\vert ^{l_{2}}\cdot \cdot \cdot \left\vert \sigma (\alpha _{2v})\right\vert ^{l_{v}}<1.\]The last relation together with \[1\leq \left\vert \func{Norm}(\alpha )\right\vert ^{l_{2}}\leq \left\vert \alpha \dprod\limits_{j=3}^{v}\sigma (\alpha _{j})\right\vert ^{l_{2}}\leq \alpha ^{l_{2}}\dprod\limits_{j=3}^{v}\left\vert \sigma (\alpha _{j})\right\vert ^{l_{j}},\]lead to a contradiction. 

---

The lemma below is also well-known, and may be found in [14] and in [15]. For a seek of completeness we give a proof of this result.

Lemma 3.3 Let \(\tau\) be a Salem numberof degree \(2d\) with non-real conjugates \(e^{\pm i2\pi \omega _{2}},\) \(...,\) \(e^{\pm i2\pi \omega _{d}},\) and let \(n\in \mathbb{N}.\)Then, \(\tau ^{n}\)is Salem number of degree \(2d,\)and \(\tau ^{n}+1/\tau ^{n}\) is a real algebraic integer greater than \(2\)whose other conjugatesare \(2\cos (2\pi n\omega _{2}),\) \(...,\) \(2\cos (2\pi n\omega _{d}),\) i.e., \(\tau ^{n}+1/\tau ^{n}\)is a Salem trace number of degree \(d.\) Moreover, the sequence \(((2\cos (2\pi n\omega _{2}),\) \(...,\) \(2\cos (2\pi n\omega _{d}))_{n\in \mathbb{N}}\)is dense in \([-2,2]^{d-1}.\)

Proof. Let \(\sigma _{1},...,\sigma _{2d}\) denote the distinct embeddings of \(\mathbb{Q}(\tau )\) into \(\mathbb{C},\) labelled so that \(\sigma _{1}(\tau )=\tau ,\) \(\sigma _{2}(\tau )=e^{i2\pi \omega _{2}},\) \(...,\) \(\sigma _{d}(\tau )=e^{i2\pi \omega _{d}},\) and \(\sigma _{d+j}(\tau )=1/\sigma _{j}(\tau )\) for all \(j\in \{1,\) \(...,\) \(d\}.\) Then, the conjugates of \(\tau ^{n},\) where \(n\in \mathbb{N},\) are the numbers \(\sigma _{1}(\tau )^{n},\) \(...,\) \(\sigma _{d}(\tau )^{n},\) and as \[\left\vert \sigma _{j}(\tau ^{n})\right\vert \leq 1<\tau ^{n}=\sigma _{1}(\tau ^{n}),\text{ \;}\forall j\in \{2,\text{ }...,\text{ }2d\},\]\(\tau ^{n}\) is a Salem number of degree \(2d.\) Also, from its definition, a Salem number is a unit, and hence \(\tau ^{n}+1/\tau ^{n}\) is an algebraic integer. Moreover, as \[\sigma _{1}(\tau ^{n}+1/\tau ^{n})=\sigma _{d+1}(\tau ^{n}+1/\tau ^{n})=\tau ^{n}+1/\tau ^{n}>2\]and \[-2<\sigma _{j}(\tau ^{n}+1/\tau ^{n})=\sigma _{d+j}(\tau ^{n}+1/\tau ^{n})=2\cos (2\pi n\omega _{j})<2,\text{ \;}\forall j\in \{2,...,d\},\]\(\tau ^{n}+1/\tau ^{n}\) is repeated twice by the action of the distinct embeddings of \(\mathbb{Q}(\tau )\) into \(\mathbb{C};\) thus \(\deg (\tau ^{n}+1/\tau ^{n})=d\) and the other conjugates of \(\tau ^{n}+1/\tau ^{n}\) are\(2\cos (2\pi n\omega _{2}),\) \(...,\) \(2\cos (2\pi n\omega _{d}).\)

To show the last assertion, fix a real number \(\varepsilon >0\) and an element \((r_{2},...,r_{d})\) of \((-2,2)^{d-1}.\) Then, Lemmas 3.2 and 3.0 give that there are integers \(p_{2},\) \(...,\) \(p_{d},\) \(n\geq 1\) such that \[\max_{2\leq j\leq d}\left\vert n\omega _{j}-p_{j}-\frac{\arccos (r_{j}/2)}{2\pi }\right\vert <\frac{\varepsilon }{4\pi }.\]Therefore, for each \(j\in \{2,...,d\},\) \[\left\vert \cos (2\pi n\omega _{j}-2\pi p_{j})-\cos (\arccos (\frac{r_{j}}{2}))\right\vert <\left\vert (2\pi n\omega _{j}-2\pi p_{j})-\arccos (\frac{r_{j}}{2})\right\vert <\frac{\varepsilon }{2},\text{ }\]\(\left\vert 2\cos (2\pi k\omega _{j})-r_{j}\right\vert <\varepsilon ,\) and so the sequence \(((2\cos (2\pi k\omega _{2}),...,2\cos (2\pi k\omega _{d}))_{k\in \mathbb{N}}\)is dense in \([-2,2]^{d-1}.\) 

---

The proofs of Theorems 1.3(i) and 1.3(iv) are based on the result below.

Lemma 3.4 Let \(K\)be a totally real number field of degree \(d\geq 2.\) Then, we have the following.

(i) If \(\theta \in \wp _{K},\)then \(\theta >2\) and so \(\theta\) is a Salem trace number of degree \(d,\) except when \(\theta =(1+\sqrt{5})/2\) (and \(K=\mathbb{Q}(\sqrt{5}));\)

(ii) the set \(\{(\sigma _{2}(\theta ),...,\sigma _{d}(\theta ))\) \(\mid \theta \in \wp _{K}\},\)where \(\sigma _{2},...,\sigma _{d}\)are the distinct embeddings of \(K\;\)into \(\mathbb{R}\)other than the identity of \(K,\)is dense in \([-1,1]^{d-1}.\)

Proof. (i) Let \(\theta \in \wp _{K}.\) Then, the other conjugates of \(\theta\) lie in the interval \((-1,1)\subset (-2,2),\) and so \(\theta\) is a Salem trace number whenever it is greater than \(2.\) In fact, a short calculation shows that \(\theta >2^{(d-1)/2},\) as stated by [17, Lemma 2], and so Lemma 3.4(i) is true, since \((1+\sqrt{5})/2\)is the unique quadratic Pisot number less than \(2.\)

(ii) Fix a Salem trace number \(\beta ,\) satisfying \(\mathbb{Q}(\beta )=K.\) Such an element exists by Lemma 3.4(i), since \(\wp _{K}\) is relatively dense in \([1,\infty ).\) Then, the first assertions in Lemmas 3.1 and 3.3 state that there is Salem number \(\tau\) of degree \(2d\) such that \(\beta =\tau +1/\tau ,\) and for each \(n\in \mathbb{N}\) the number \(\beta _{n}:=\tau ^{n}+1/\tau ^{n}\) is a Salem trace number of degree \(d.\) Also, the last assertions in Lemmas 3.1 and 3.3 give that \(\beta _{n}\in \mathbb{Q}(\beta )\) and the set \(\{(\sigma _{2}(\beta _{n}),...,\sigma _{d}(\beta _{n}))\mid n\in \mathbb{N}\}\) is dense in \([-2,2]^{d-1}.\) It follows that the set \[N:=\{n\in \mathbb{N}\mid (\sigma _{2}(\beta _{n}),...,\sigma _{d}(\beta _{n}))\in (-1,1)^{d-1}\},\]is not finite, and the sequence \((\sigma _{2}(\beta _{n}),...,\sigma _{d}(\beta _{n}))_{n\in N}\) is dense in \([-1,1]^{d-1}.\) Hence, the set \(\{(\sigma _{2}(\theta ),...,\sigma _{d}(\theta ))\) \(\mid \theta \in \wp _{K}\},\) containing \(\{(\sigma _{2}(\beta _{n}),...,\sigma _{d}(\beta _{n}))\mid n\in N\},\) is also dense in \([-1,1]^{d-1},\) since \(\beta _{n}\in \wp _{K}\) for each \(n\in N.\) 

---

4. Proof of Theorem 1.1 and Corollary 1.2

As mentioned in the introduction, Theorem 1.1 is true when \(K=\mathbb{Q}.\) Suppose that the degree \(d\) of the real number field \(K\) is greater than \(1,\) and let \(\sigma _{1},...,\sigma _{d}\) be the distinct embeddings of \(K\) into \(\mathbb{C},\) labelled so that \(\sigma _{1}\) is the identity of \(K,\) \(\sigma _{1},...,\sigma _{r}\) are real, \(\sigma _{r+1},...,\sigma _{r+s}\) are non-real, and \(\sigma _{r+s+1}=c\circ \sigma _{r+1},\) \(...,\) \(\sigma _{r+2s}=c\circ \sigma _{r+s},\) where \(c\) is the complex conjugation, \(\circ\) is the composition function, and \(d=r+2s.\)

Notice that if \((\alpha ,\alpha ^{\prime })\in \wp _{K}^{2},\) then \(\alpha \alpha ^{\prime }\in \wp _{K},\) since \[\left\vert \sigma _{j}(\alpha \alpha ^{\prime })\right\vert =\left\vert \sigma _{j}(\alpha )\right\vert \left\vert \sigma _{j}(\alpha ^{\prime })\right\vert <1<\alpha \alpha ^{\prime }=\sigma _{1}(\alpha \alpha ^{\prime }),\text{ }\forall \text{ }j\in \{2,...,d\},\]and \(\alpha \alpha ^{\prime }\) is not repeated under the action of the distinct embeddings of \(K\) into \(\mathbb{C};\) thus \(\wp _{K}\) is closed under multiplication, and the powers of any element of \(\wp _{K}\) belong to \(\wp _{K}\) too.

(i) Let \(\alpha \in \wp _{K}\) with conjugates \(\alpha _{1}:=\sigma _{1}(\alpha ),...,\alpha _{d}:=\sigma _{d}(\alpha ).\) By replacing, if necessary, \(\alpha\) by \(\alpha ^{2}\) we may assume that \(\alpha _{1}>0,\) \(...,\) \(\alpha _{r}>0.\) In fact, we shall prove that there are infinitely many natural numbers \(q\) such that

\[\alpha _{q}^{\prime }:=\alpha ^{q}-1\in \wp _{K}.\]This implies that \(1=\alpha ^{q}-\alpha _{q}^{\prime }\in D_{K},\) and \(1\) may be expressed as a difference of two elements of \(\wp _{K}\) in infinitely many ways.

In fact, if \(s=0,\) then \(\alpha _{q}^{\prime }\in \wp _{K},\) \(\forall\) \(q\in \mathbb{N},\) because the relations \(\sigma _{j}(\alpha ^{q}-1)=\alpha _{j}^{q}-1\in (-1,0),\) when \(j\) runs through \(\{2,...,d\},\) together with the fact that the norm of \(\alpha _{q}^{\prime }\) is a non-zero integer imply the inequality \(\alpha _{q}^{\prime }>1.\)

Now suppose \(s\geq 1,\) and set \(\alpha _{r+1}:=\rho _{1}e^{i2\pi \omega _{1}},\) \(...,\) \(\alpha _{r+s}:=\rho _{s}e^{i2\pi \omega _{s}}.\) Then, \(\omega _{j}\notin \mathbb{Q},\) \(\forall\) \(j\in \{1,...,s\},\) since otherwise there are natural numbers \(n\) and \(j\leq s\) such that \(\alpha _{r+j}^{n}=\rho _{j}^{n}e^{i2\pi n\omega _{j}}=\rho _{j}^{n}\in \mathbb{R},\) and so \(\mathbb{Q}(\alpha _{r+j}^{n})\varsubsetneq \mathbb{Q}(\alpha _{r+j})\) contradicting the fact that \(\mathbb{Q}(\alpha ^{n})=\mathbb{Q}(\alpha )\) (in fact Lemma 3.2 yields also \(\omega _{j}\notin \mathbb{Q},\) \(\forall\) \(j\in \{1,...,s\}).\)

It follows by Dirichlet’s approximation theorem (see for instance [3]) that for each real number \(Q\geq 6^{s},\) there is a natural number \(q\leq Q\) and \(s\) integers \(p_{1},\) \(...,\) \(p_{s\text{ }}\) such that \[0<\left\vert q\omega _{j}-p_{j}\right\vert <\frac{1}{Q^{1/s}}\leq \frac{1}{6},\text{ }\forall \text{ }j\in \{1,...,s\}.\]Hence, \(\left\vert 2\pi q\omega _{j}-2\pi p_{j}\right\vert <\) \(\pi /3,\) \(\cos (2\pi q\omega _{j})>1/2,\) \[\left\vert \alpha _{r+j}^{q}-1\right\vert ^{2}=(\rho _{j}^{q}e^{i2\pi q\omega _{j}}-1)(\rho _{j}^{q}e^{-i2\pi q\omega _{j}}-1)=\rho _{j}^{q}(\rho _{j}^{q}-2\cos (2\pi q\omega _{j}))+1<1,\]and \(\left\vert \alpha _{r+j}^{q}-1\right\vert <1,\) \(\forall\) \(j\in \{1,...,s\}.\)

Because \(\sigma _{j}(\alpha ^{q}-1)=\alpha _{j}^{q}-1\) \(\in (-1,0),\) \(\forall\) \(j\in \{2,...,r\},\) whenever \(r\geq 2,\) \(\left\vert \sigma _{r+s+j}(\alpha ^{q}-1)\right\vert =\left\vert c(\alpha _{r+j}^{q}-1)\right\vert =\left\vert \alpha _{r+j}^{q}-1\right\vert <1\) for all \(j\in \{1,...,s\},\) and the norm of the positive algebraic integer \(\alpha ^{q}-1\) is a non-zero integer, we see that \(\alpha ^{q}-1\in \wp _{K}\) and so Theorem 1.1(i) is true. Since \(n\omega _{j}-p_{j}\neq 0\) for all \((j,n,p_{j})\in \{1,...,s\}\times \mathbb{N}\times \mathbb{Z},\) by letting \(Q\) tend to infinity we obtain that there are infinitely many \(q\in \mathbb{N}\) such that \(\alpha ^{q}-1\in \wp _{K}.\)

(ii) and Corollary 1.2 Let \(\tau\) be a Pisot number or a Salem number, satisfying \(K=\mathbb{Q}(\tau ).\) Then, Theorem 1.1(i) gives that \(1=\alpha -\alpha ^{\prime }\) for some \((\alpha ,\alpha ^{\prime })\in \wp _{K}^{2},\) and so \(\tau =\tau \alpha -\tau \alpha ^{\prime }.\) Arguing as above, we get from the relations \(\left\vert \sigma _{j}(\tau \alpha )\right\vert =\left\vert \sigma _{j}(\tau )\sigma _{j}(\alpha )\right\vert \leq \left\vert \sigma _{j}(\alpha )\right\vert <1\) and \(\left\vert \sigma _{j}(\tau \alpha ^{\prime })\right\vert =\left\vert \sigma _{j}(\tau )\sigma _{j}(\alpha ^{\prime })\right\vert \leq \left\vert \sigma _{j}(\alpha ^{\prime })\right\vert <1,\) \(\forall\) \(j\) \(\in \{2,...,d\},\) that the positive algebraic integers \(\tau \alpha\) and \(\tau \alpha ^{\prime }\) belong to \(\wp _{K},\) and so \(\tau \in D_{K}.\) In particular, \(\wp _{K}\subset D_{K},\) and \(\wp _{K}\varsubsetneq D_{K},\) as \(1\in D_{K}.\)

(iii) The proof of the second assertion is immediate. Indeed, if \(\func{card}(\tciFourier _{K})=1,\) then (2) gives that for any \(n\in \mathbb{N},\) \(\theta _{n}-\theta _{1}=(n-1)d_{1},\) \[(n-1)\left\vert \sigma _{j}(d_{1})\right\vert =\left\vert \sigma _{j}(\theta _{n}-\theta _{1})\right\vert \leq \left\vert \sigma _{j}(\theta _{n})\right\vert +\left\vert \sigma _{j}(\theta _{1})\right\vert \leq 2,\]where \(j\) is any fixed element of \(\{2,...,d\},\) and this last relation is not true when \(n\) sufficiently is large.

To show the first assertion, recall from the proof of Theorem 1.1(i) that if \(\alpha \in \wp _{K}\) and the real conjugates of \(\alpha\) are positive, then there are infinitely many natural numbers \(q\) such that \(\alpha ^{q}-1\in \wp _{K}.\) If we apply this result, when \(r\geq 2,\) to the square of an element \(\alpha\) of \(\wp _{K},\) we see that each element of \(D_{K}\) of the form \((\alpha ^{2q}-1)-\alpha ^{q}\) does belong to \(\wp _{K},\) as \(\sigma _{2}(\alpha )^{2q}-1-\sigma _{2}(\alpha )^{q}<-1,\) and so Theorem 1.1(iii) is true, whenever \(r\geq 2.\)

Now, suppose that all other conjugates of \(\alpha\) are non-real, and we claim there are infinitely many natural numbers \(n\) such that \(\alpha ^{n}+1\in \wp _{K}.\) Indeed, setting \(\sigma _{2}(\alpha ):=\rho _{1}e^{i2\pi \omega _{1}},\) \(...,\) \(\sigma _{1+s}(\alpha ):=\rho _{s}e^{i2\pi \omega _{s}},\) we have by Lemma 3.2 that \(1,\)\(2\omega _{1},\) \(...,\) \(2\omega _{s}\)are\(\mathbb{Q}\)-linearly independent, and so it follows by Lemma 3.0 that for each \(\varepsilon \in (0,1/6)\) there are integers \(p_{1},\) \(...,\) \(p_{s}\) and a natural number \(n\) such that \[\max_{1\leq j\leq s}\left\vert n\omega _{j}-p_{j}-\frac{1}{2}\right\vert <\varepsilon <\frac{1}{6}.\]Hence, for each \(j\in \{1,...,s\},\) \(\left\vert 2n\pi \omega _{j}-2\pi p_{j}-\pi \right\vert <\pi /3,\) \(\cos (2\pi n\omega _{j})<-1/2,\) and so \[\left\vert \sigma _{1+j}(\alpha )^{n}+1\right\vert ^{2}=(\rho _{j}^{n}e^{i2\pi n\omega _{j}}+1)(\rho _{j}^{n}e^{-i2\pi n\omega _{j}}+1)=\rho _{j}^{q}(\rho _{j}^{q}+2\cos (2\pi q\omega _{j}))+1<1,\]and in the same as in the end of the proof of Theorem 1.1(i), we get that \(\alpha ^{n}+1\in \wp _{K}.\) Also, since \(n\omega _{j}-p_{j}-\frac{1}{2}\neq 0\) for all \((j,n,p_{j})\in \{1,...,s\}\times \mathbb{N}\times \mathbb{Z},\) by letting \(\varepsilon\) tend to zero we get that there are infinitely many \(n\) such that \(\alpha ^{n}+1\in \wp _{K}\) and this ends the proof of the claim.

We can now easily deduce that Theorem 1.1(iii) is true. Indeed, choose a large integer \(q\) so that \(\alpha ^{q}-1\in \wp _{K}\) and \(\left\vert \sigma _{2}(\alpha )\right\vert ^{q}<1/2,\) and let \(n\) be any integer satisfying \(\alpha ^{n}+1\in \wp _{K}\) and \(n>q.\) Then, any element \(\beta _{n}:=(\alpha ^{n}+1)-(\alpha ^{q}-1)\) of \(D_{K},\) satisfies \(\left\vert \sigma _{2}(\beta _{n})\right\vert =\left\vert 2+\alpha ^{n}-\alpha ^{q}\right\vert >2-\left\vert \sigma _{2}(\alpha )\right\vert ^{n}-\left\vert \sigma _{2}(\alpha )\right\vert ^{q}>2-2\left\vert \sigma _{2}(\alpha )\right\vert ^{q}>1,\) and so \(\beta _{n}\notin \wp _{K}.\) 

---

5. Proof of Theorem 1.3

Let \(K\) be totally real number field of degree \(d\geq 2,\) and let \(\sigma _{1},\) \(...,\) \(\sigma _{d}\) be the distinct embeddings of \(K\) into \(\mathbb{R},\) where \(\sigma _{1}\) is the identity of \(K.\)

(i) As mentioned in the relation (1), \(D_{K}\) is always contained in \(E_{K}.\) To show the inclusion \(E_{K}\subset\) \(D_{K},\) consider an element \(\beta\) of \(E_{K}.\) Then, \(\beta\) is a positive algebraic integer of \(K\) such that \((\sigma _{2}(\beta ),...,\sigma _{d}(\beta ))\in (-2,2)^{d-1}.\) Recall, by Lemma 3.4(ii), that the set \(\{(\sigma _{2}(\theta ),...,\sigma _{d}(\theta ))\) \(\mid \theta \in \wp _{K}\}\)is dense in \([-1,1]^{d-1},\) and hence we can find (infinitely many) \(\theta \in \wp _{K}\) such that for each \(j\in \{2,...,d\},\) \[-1<\sigma _{j}(\theta )<1-\sigma _{j}(\beta )\text{ \;if }\sigma _{j}(\beta )>0,\]and \[-1-\sigma _{j}(\beta )<\sigma _{j}(\theta )<1\text{\;\;when }\sigma _{j}(\beta )<0.\]It follows that the real algebraic integer \(\theta ^{\prime }:=\beta +\theta >1\) belongs to \(K\) and satisfies \(\sigma _{j}(\theta ^{\prime })=\sigma _{j}(\beta )+\sigma _{j}(\theta )\in (-1,1)\) for all \(j\in \{2,...,d\};\) thus \(\deg (\theta ^{\prime })=d,\) as it is not repeated under the action of the distinct embeddings of \(K\) into \(\mathbb{R},\) \(\theta ^{\prime }\in \wp _{K}\) and \(\beta =\theta ^{\prime }-\theta \in D_{K}.\)

(ii) From the definition of a root of unity trace number we have that \(U_{K}\subset E_{K}\cap (0,2).\) To prove the inclusion inverse consider an algebraic integer \(\beta\) of \(K\) such that \((\sigma _{1}(\beta ),\sigma _{2}(\beta ),...,\sigma _{d}(\beta ))\in (0,2)\times (-2,2)\times \cdot \cdot \cdot \times (-2,2).\) Because the zeros of the monic polynomial \[P(x):=(x^{2}-\sigma _{1}(\beta )x+1)\cdot \cdot \cdot (x^{2}-\sigma _{d}(\beta )x+1)\in \mathbb{Z}[x]\] are all non-real numbers of modulus \(1,\) we have, by Kronecker’s theorem (see for instance [3]), that these zeros are roots of unity. Hence, there are natural numbers \(n\) and \(k\leq n\) with \(\gcd (k,n)=1\) such that \(\beta =\sigma _{1}(\beta )=e^{i2\pi k/n}+1/e^{i2\pi k/n}=2\cos (2k\pi /n).\) Further, the condition \(\beta \in (0,2)\) implies that \(n\geq 5,\) \(k\leq n-1\) and \(\beta \in U_{K};\) thus \(U_{K}=E_{K}\cap (0,2),\) and so, by Theorem 3.1(i), \(U_{K}=D_{K}\cap (0,2).\)

(iii) The first assertion follows from Theorems 1.3(i) and 1.3(ii). It is also clear that any element of the set \(\{\beta \in \mathbb{Z}_{K}\cap (2,\infty )\mid (\sigma _{2}(\beta ),...,\sigma _{d}(\beta ))\in (-2,2)^{d-1}\}\) is not repeated under the action of the distinct embeddings of \(K\) into \(\mathbb{R}\) and so it is of degree \(d.\)

(iv) Fix an element \((\varepsilon _{2},...,\varepsilon _{d})\) of the set \(\{-1,1\}^{d-1}.\) Then, Lemma 3.4(ii) gives that there is a subsequence \((\theta _{f(n)})_{n\in \mathbb{N}}\) of the sequence \((\theta _{n})_{n\in \mathbb{N}}\) such that \(\lim_{n\rightarrow \infty }(\sigma _{2}(\theta _{f(n)}),...,\sigma _{d}(\theta _{f(n)}))=(\varepsilon _{2},...,\varepsilon _{d}).\) Because \(\{\theta _{f(n)+1}-\theta _{f(n)}\mid\) \(n\in \mathbb{N\}}\) is contained in the finite set \(\tciFourier _{K},\) there exist an element \(d_{\ast }\) of \(\tciFourier _{K}\) and an infinite subset \(N\) of \(\mathbb{N}\) such that \(\theta _{f(n)+1}-\theta _{f(n)}=d_{\ast }\) for all \(n\in N.\) Therefore for each \((j,n)\in \{2,...,d\}\times N,\) \(\sigma _{j}(\theta _{f(n)})+\sigma _{j}(d_{\ast })=\sigma _{j}(\theta _{f(n)+1}),\) and hence \[-1<\sigma _{j}(\theta _{f(n)})+\sigma _{j}(d_{\ast })<1.\]Letting the element \(n\) of \(N\) tend to infinity, we get from the last relation that \(-1-\varepsilon _{j}\leq \sigma _{j}(d_{\ast })\leq 1-\varepsilon _{j},\) and so \(\sigma _{j}(d_{\ast })<0\) (resp. \(\sigma _{j}(d_{\ast })>0)\) when \(\varepsilon _{j}=1\) (resp. \(\varepsilon _{j}=-1).\)

Therefore, for any \((\varepsilon _{2},...,\varepsilon _{d})\in \{-1,1\}^{d-1},\) there is \(d_{\ast }\in \tciFourier _{K}\) such that each \((j,n)\in \{2,...,d\},\) the numbers \(\sigma _{j}(\beta )\) and \(\varepsilon _{j}\) have opposite signs, and so there are \(2^{d-1}\) distinct elements of the set \(\tciFourier _{K},\) which are in a one-to-one correspondence with the elements \(\{-1,1\}^{d-1}.\)

(v) Since \(\tciFourier _{K}\cap (0,2)\subset D_{K}\cap (0,2),\) the first assertion follows from Theorem 1.3(ii). Also, Theorem 1.3(ii) says that small elements of \(D_{K}\) belong to \(U_{K}.\) In particular, \(\min D_{K}=\min U_{K}=\) \(u_{1},\) and so \(u_{1}\leq \min \{1,d_{1}\},\) since \(\{d_{1},1\}\subset D_{K}.\) Using (2), we get \(u_{1}=l_{1}d_{1}+\cdot \cdot \cdot +l_{\func{card}(\tciFourier _{K})}d_{\func{card}(\tciFourier _{K})}\geq (l_{1}+\cdot \cdot \cdot +l_{\func{card}(\tciFourier _{K})})d_{1}\geq d_{1},\) for some non-negative integers \(l_{1},\) \(...,\) \(l_{\func{card}(\tciFourier _{K})},\) and hence \(u_{1}=d_{1}\leq 1.\)

Now, suppose \(Card(U_{K})\geq 2.\) Then, \(\min (D_{K}\backslash \{u_{1}\})=\) \(u_{2},\) and \(u_{2}\leq d_{2},\) as \(d_{2}\in D_{K}\backslash \{d_{1}\}=D_{K}\backslash \{u_{1}\}.\) Also, writing \(u_{2}=l_{1}d_{1}+\cdot \cdot \cdot +l_{\func{card}(\tciFourier _{K})}d_{\func{card}(\tciFourier _{K})},\) for some non-negative integers \(l_{1},\) \(...,\) \(l_{\func{card}(\tciFourier _{K})},\) we obtain by the relation \(u_{2}\leq d_{2}\) that \(u_{2}=d_{2}\) or \(u_{2}=l_{1}d_{1}\) from some natural \(l_{1}\geq 2.\) In fact this last possibility does not hold because the equalities \(u_{2}=l_{1}d_{1}=l_{1}u_{1}\) imply that the conjugates of the non-zero algebraic integer \(u_{1}=u_{2}/l_{1}\) belong to \((-1,1);\) thus \(u_{2}=d_{2}\) (in the same way we can show that \(u_{3}=d_{3},\) whenever \(Card(U_{K})\geq 3\) and \(u_{3}\) is not a linear combination of \(u_{1}\) and \(u_{2}\) with non-negative integer coefficients).

(vi) Suppose \(Card(U_{K})=1.\) Then,\(U_{K}=\{1\},\)as \(1\in U_{K}.\) It follows by Theorems 1.3(v) and 1.3(iii), respectively, that \(d_{1}=1\)and \(d_{2}\in T_{K};\) thus \(\min T_{K}\leq d_{2}.\) In the same way as in the last proof, we get \(\min T_{K}=d_{2}\) or \(\min T_{K}=l_{1}d_{1}=l_{1}\in \mathbb{N}.\) Because the degree of each element of \(T_{K}\) is equal to \(d,\) as asserted by Theorem 1.3(iii), this last possibility does not hold, and so \(\min T_{K}=d_{2}.\) 

---

In a private communication and after having finished writing this manuscript, I was informed by Dubickas that he considered in [6] the more general problem about algebraic integers which may written as a difference of two Pisot numbers. Using another approach, he obtained certains results improving some ones of the present manuscript.

References

[1] M. J. Bertin, A. Decomps-Guilloux, M. Grandet-Hugo, M. Pathiaux-Delefosse and J. P. Schreiber, Pisot and Salem numbers, Birkhäuser Verlag Basel, 1992.

[2] M. J. Bertin and T. Zaı̈mi, Complexes Pisot numbers in algebraic number fields, C. R. Math., Acad. Sci. Paris 353, No. 11 (2015), 965-967.

[3] Y. Bugeaud, Distribution modulo one and diophantine approximation, Cambridge university press, 2012.

[4] J. W. S. Cassels, An introduction to Diophantine Approximation, Cambridge University Press, 1952.

[5] A. Dubickas, Every Salem number is a difference of two Pisot numbers, Proc. Edinb. Math. Soc., II. Ser. 66, No. 3 (2023), 862-867.

[6] A. Dubickas, Every Salem number is a difference of two Pisot numbers, (preprint).

[7] A. H. Fan and J. Schmeling, \(\varepsilon -\)Pisot numbers in any real algebraic number field are relatively dense, J. Algebra 272 (2004), 470-475.

[8] J. C. Lagarias, Meyer’s concept of quasicrystal and quasiregular sets, Commun. Math. Phys. 179 (1996), 365-376.

[9] L. Kronecker, Zwei Sätze über Gleichungen mit ganzzahligen Coefficienten, J. Reine Angew. Math., 53 (1857), 173-175

[10] Y. Meyer, Nombres de Pisot, nombres de Salem et analyse harmonique, Lecture Notes in Mathematics, Springer-Verlag, 117 (1970).

[11] M. Mignotte, sur les conjugués des nombres de Pisot, C. R. Acad. Sci. Paris, 298 (1984), 21-24.

[12] R. V. Moody, Meyer sets and their duals, The Mathematics of long-range aperiodic order (Waterloo, ON, 1995), 403-441, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 489, Kluwer Acad. Publ., Dordrecht, 1997.

[13] C. Pisot, Quelques aspects de la théorie des entiers algébriques, Séminaire de mathématiques supérieures, Université de Montréal, 1963.

[14] R.Salem, Algebraic numbers and Fourier analysis, D. C. Heath and Co., Boston, Mass., 1963.

[15] C. J. Smyth, Seventy years of Salem numbers, Bull. Lond. Math. Soc. 47, No. 3, 379-395 (2015).

[16] THE PARI GROUP, PARI/GP version 2.5.1, 2012, avaible from

http://pari.math.u-bordeaux.fr/.

[17] T. Zaı̈mi, Sur les nombres de Pisot totalement reels, Arab J. Math. Sci 5, No. 2 (1999), 19-32.

] T. Zaı̈mi, On \(\varepsilon -\)Pisot numbers, New York J. Math. 15 (2009), 415-422.

[19] T. Zaı̈mi, Commentaires sur quelques résultats sur les nombres de Pisot, J. Théor. Nombres Bordx. 22, No. 2 (2010), 513-524.

[20] T. Zaı̈mi, Comments on Salem polynomials, Arch. Math. 117 (2021), 41–51.

Department of Mathematics and Statistics College of Science

Imam Mohammad Ibn Saud Islamic University (IMSIU)

P. O. Box 90950

Riyadh 11623 Saudi Arabia

Email: tmzaemi@imamu.edu.sa