November 28, 2023
We use bounds on bilinear forms with Kloosterman fractions and improve the error term in the asymptotic formula of Balazard and Martin (2023) on the average value of the smallest denominators of rational numbers in short intervals.
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Given integer \(N \geqslant 1\), and \(j =1, \ldots, N\), we denote by \(q_j(N)\) the smallest integer \(q\) such that for some \(a\) we have \[\frac{a}{q} \in \left( \frac{j-1}{N}, \frac{j}{N}\right].\]
Next, we consider the average value \[S(N) = \frac{1}{N} \sum_{j=1}^N q_j(N).\]
Recently, Balazard and Martin [1] have confirmed the conjecture of Kruyswijk and Meijer [2] that \[S(N) \sim \frac{16}{\pi^2} N^{3/2}\] and in fact established the following much more precise asymptotic formula \[\label{eq:S32asymp} S(N) = \frac{16}{\pi^2} N^{3/2} + O\left( N^{4/3} (\log N)^2\right),\tag{1}\] see [1]. Note that the asymptotic formula (1 ) improves on previous upper and lower bounds of Kruyswijk and Meijer [2] and Stewart [3], for example on the previous inequalities \[1.35N^{3/2} < S(N) < 2.04 N^{3/2}\] in [3] (note that \(16/\pi^2 = 1.6211 \ldots\)). For other related results, see [4]–[9] and references therein.
The bound on the error term in (1 ) is based on the classical bound of Kloosterman sums, see, for example, [10].
Here, we use bounds on bilinear forms with Kloosterman fractions due to Duke, Friedlander and Iwaniec [11] and improve the error term in the asymptotic formula (1 ) as follows.
Theorem 1. We have \[S(N) = \frac{16}{\pi^2} N^{3/2} + O\left( N^{29/22 + o(1)} \right),\] as \(N\to \infty\).
As usual, we use the expressions \(U \ll V\) and \(U=O(V)\) to mean \(|U|\leqslant c V\) for some constant \(c>0\) which throughout this paper is absolute.
We have \[\label{eq:S32and32R} S(N) = \frac{16}{\pi^2} N^{3/2} + R(N),\tag{2}\] where by [1] we can write \[\label{eq:R32and32T} R(N) \ll T_{11}(N) + T_{12}(N) + T_{2}(N)\tag{3}\] for some quantities \(T_{11}(N)\), \(T_{12}(N)\) and \(T_{2}(N)\) which are estimated in [1] separately. In particular, by [1] we have \[\label{eq:T1232and32T2} T_{12}(N) \ll N^{5/4} (\log N)^2 \qquad and \qquad T_{2}(N) \ll N^{5/4} (\log N)^2.\tag{4}\] Therefore. the error term in (1 ) comes from the bound \[\label{eq:T11} T_{11}(N) \ll N^{4/3} (\log N)^2\tag{5}\] given by [1].
We now see from (2 ), (3 ) and (4 ), that in order to establish Theorem 1 we only need to improve (5 ) as \[\label{eq:T11-new} T_{11}(N) \ll N^{29/22 + o(1)} .\tag{6}\]
We first recall the following expression for \(T_{11}(N)\) given in [1]: \[\label{eq:T11-B} T_{11}(N) = \sum_{s \geqslant\sqrt{N} } \sum_{\substack{1 \leqslant r \leqslant R_s\\\gcd(r,s)=1}} r B_1\left(\frac{N r^{-1}}{s}\right)\tag{7}\] with the Bernoulli function \[B_1(u) = \begin{cases} 0, & \text{if}\;u \in {\mathbb{Z}}, \\ \{u\}-1/2, & \text{if}\;u \in {\mathbb{Z}}, \end{cases}\] where \(\{u\}\) is the fractional part of a real \(u\), the inversion \(r^{-1}\) in the fractional part \(\{N r^{-1}/s\}\) is computed modulo \(s\) and \(R_s\) is a certain sequence of positive integers, satisfying \[\label{eq:Rs} R_s \ll N/s\tag{8}\] (we refer to [1] for an exact definition, which is not important for our argument).
It is more convenient for us to work with the function \[\psi(u) = \{u\}-1/2,\] which coincides with \(B_1(u)\) for all \(u \not \in {\mathbb{Z}}\).
In particular, \[B_1\left(\frac{N r^{-1}}{s}\right)= \psi\left(\frac{N r^{-1}}{s}\right)\] unless \(s \mid N\).
Using the classical bound on the divisor function \[\label{eq:Div} \tau(k) = k^{o(1)},\tag{9}\] for an integer positive \(k \to \infty\) (see, for example, [10]), we infer from (7 ) that \[\label{eq:T11-psi} T_{11}(N) =U(N) + E(N) ,\tag{10}\] where \[\label{eq:UN} U(N) = \sum_{s \geqslant\sqrt{N} } \sum_{\substack{1 \leqslant r \leqslant R_s\\\gcd(r,s)=1}} r \psi\left(\frac{N r^{-1}}{s}\right),\tag{11}\] and, using (8 ), \[\label{eq:EN} E(N) \ll \sum_{\substack{s \geqslant\sqrt{N} \\s \mid N}} R_s^2 \ll N^2 \sum_{\substack{s \geqslant\sqrt{N} \\s \mid N}} s^{-2} \leqslant N^{1+o(1)}.\tag{12}\]
By a result of Vaaler [12], see also [13] we have the following approximation to \(\psi(u)\).
Lemma 2. For any integer \(H\geqslant 1\) there is a trigonometric polynomial \[\psi_H(u) = \sum_{1\leqslant\left| h \right| \leqslant H} \frac{a_h}{-2i\pi h} {\mathbf{\,e}}(hu)\] for coefficients \(a_h\in[0,1]\) and such that \[\label{eq:Vaal32Ined} \left| \psi(u)-\psi_H(u) \right| \leqslant\frac{1}{2H+2} \sum_{\left| h \right|\leqslant H} \left(1-\frac{\left| h \right|}{H+1}\right) {\mathbf{\,e}}(hu).\qquad{(1)}\]
For an integer \(q\), let \({\mathbf{\,e}}(z) = \exp(2 \pi z)\). Here we collect some estimates on bilinear form with exponantials \({\mathbf{\,e}}\left(hr^{-1}/s\right)\) where, as before, \(r^{-1}\) in the argument is computed modulo \(s\).
For \(U\geqslant 1\) we aslo also use \(u \sim U\) to indicate \(U \leqslant u < 2U\).
We start with recalling the following bound of Duke, Friedlander and Iwaniec [11].
Lemma 3. For sequences \(\boldsymbol{\alpha}= \{\alpha_r\}_{r=1}^\infty\), \(\boldsymbol{\beta}= \{\beta_s\}_{s=1}^\infty\) of complex numbers, an nonzero integer \(K\) and real positive \(R\) and \(S\) we have \[\begin{align} & \left| \sum_{s \sim S} \sum_{\substack{r \sim R\\\gcd(r,s) = 1}} \alpha_r \beta_s {\mathbf{\,e}}\left(Kr^{-1}/s\right) \right|\\ &\qquad \quad \leqslant\|\boldsymbol{\alpha}\| \|\boldsymbol{\beta}\| \left((R + S)^{1/2} +\left(1 + \frac{K}{RS}\right)^{1/2} \min\{R,S\}\right) (RS)^{o(1)}, \end{align}\] where \[\|\boldsymbol{\alpha}\| =\left( \sum_{r \sim R}|\alpha_r|^2\right)^{1/2} \qquad and \qquad\|\boldsymbol{\beta}\| =\left( \sum_{s \sim S}|\beta_s|^2\right)^{1/2}.\]
Next, given two sequences of complex numbers \[\boldsymbol{\alpha}= \{\alpha_r\}_{r=1}^\infty \qquad and \qquad\boldsymbol{\beta}= \{\beta_s\}_{s=1}^\infty,\] a sequence of positive integers \[{\mathcal{R}}= \{\beta_s\}_{s=1}^\infty\] and an integer \(h\), for \(S \geqslant 1\) we define the bilinear form \[\mathscr{B}_K(S; {\mathcal{R}}, \boldsymbol{\alpha}, \boldsymbol{\beta}) = \sum_{s \sim S} \sum_{\substack{r =1\\\gcd(r,s) = 1}}^{R_s} \alpha_r \beta_s {\mathbf{\,e}}\left(Kr^{-1}/s\right).\] Note that in the sums \(\mathscr{B}_K(S; {\mathcal{R}}, \boldsymbol{\alpha}, \boldsymbol{\beta})\) the range of summation over \(r\) depends on \(s\) and hence Lemma 3 does not directly apply.
We observe that for \[\label{eq:Special32case} \alpha_r = r, \quad \beta_s \ll 1, \quad R_s \ll \min\{N/s, s\}, \qquad r,s =1, 2, \ldots,\tag{13}\] the argument in [1] (in which we also inject the bound (9 )) immediately implies that for \[0 < |K| = N^{O(1)} \qquad and \qquad 0 < S \ll N\] we have \[\label{eq:Bound-Weil} \begin{align} \mathscr{B}_K(S; {\mathcal{R}}, \boldsymbol{\alpha}, \boldsymbol{\beta}) & \ll \sum_{s \sim S} \gcd(K, s)^{1/2} R_s s^{1/2} \log s\\ & \leqslant N^{1+o(1)}\sum_{s \sim S} \gcd(K, s)^{1/2} s^{-1/2} \\ & \leqslant N^{1+o(1)} S^{-1/2} \sum_{d\mid K} d^{1/2} \sum_{\substack{s \leqslant 2S\\ d \mid s}} 1 \\ & \leqslant N^{1+o(1)} S^{-1/2} \sum_{d\mid K} d^{1/2} \left\lfloor 2S/d\right\rfloor \\ & \leqslant N^{1+o(1)} S^{1/2} \sum_{d\mid K} d^{-1/2} \\ & \leqslant N^{1+o(1)} S^{1/2}. \end{align}\tag{14}\] Note that one can also derive (14 ) via [11] and partial summation.
In fact using the bound (14 ) for \(S \leqslant N^{2/3}\) and the trivial bound \[\mathscr{B}_K(S; {\mathcal{R}}, \boldsymbol{\alpha}, \boldsymbol{\beta}) \ll \sum_{s \sim S} R_s^2 \ll N^2S^{-1}\] in our argument below, one recovers the asymptotic formula (1 ). However using some other bounds we achieve a stronger result.
We also remark that for us only the choice of \(\boldsymbol{\alpha}= \{\alpha_r\}_{r=1}^\infty\) satisfying (13 ) matter. However we present the below results for a more general \(\boldsymbol{\alpha}\) (but still they admit even more general forms).
Using Lemma 3 together with the standard completing technique, see, for example, [10], we derive our main technical tool.
Lemma 4. For sequences \(\boldsymbol{\alpha}= \{\alpha_r\}_{r=1}^\infty\), \(\boldsymbol{\beta}= \{\beta_s\}_{s=1}^\infty\) and \({\mathcal{R}}= \{R_s\}_{s=1}^\infty\), an nonzero integer \(K\) and real \(S\) with \[\alpha_r \ll A, \quad \beta_s \ll B, \quad R_s \ll \min\{N/s, s\}, \qquad r,s =1, 2, \ldots,\] and \[N^{1/2} \ll S \ll N,\] we have \[\left| \mathscr{B}_K(S; {\mathcal{R}}, \boldsymbol{\alpha}, \boldsymbol{\beta}) \right| \leqslant AB (RS)^{1/2} \left(S^{1/2} + R+ K^{1/2} S^{-1/2} R^{1/2}\right)N^{o(1)},\] where \[R = \max\{R_s: ~ s\sim S\}.\]
Proof. Note that \[\label{eq:R32N32S} R \ll N/S \ll S.\tag{15}\]
Using the orthogonality of exponential functions, we write \[\begin{align} \mathscr{B}_K(S&; {\mathcal{R}}, \boldsymbol{\alpha}, \boldsymbol{\beta}) \\ & = \sum_{s \sim S} \sum_{\substack{r =1\\\gcd(r,s) = 1}}^{R_s} \alpha_r \beta_s {\mathbf{\,e}}\left(Kr^{-1}/s\right)\\ & = \sum_{s \sim S} \sum_{\substack{r =1\\\gcd(r,s) = 1}}^{R} \alpha_r \beta_s {\mathbf{\,e}}\left(Kr^{-1}/s\right) \frac{1}{R} \sum_{u=0}^{R-1}\sum_{t=1}^{R_s} {\mathbf{\,e}}(u(t - r)/R) \\ & = \frac{1}{R} \sum_{u=0}^{R-1} \sum_{s \sim S} \sum_{\substack{r =1\\\gcd(r,s) = 1}}^{R} \alpha_r {\mathbf{\,e}}(-ur/R) \beta_s {\mathbf{\,e}}\left(Kr^{-1}/s\right) \sum_{t=1}^{R_s} {\mathbf{\,e}}(ut/R) . \end{align}\] Using that \[\sum_{t=1}^{R_s} {\mathbf{\,e}}(ut/R) \ll \frac{R}{\min\{u, R- u\} + 1},\] see [10], we derive \[\begin{align} \mathscr{B}_K(S; {\mathcal{R}}, \boldsymbol{\alpha}, \boldsymbol{\beta}) \ll \frac{1}{R} & \sum_{u=0}^{R-1} \frac{R}{\min\{u, R- u\} + 1}\\ &\quad \times \left| \sum_{s \sim S} \sum_{\substack{r =1\\\gcd(r,s) = 1}}^{R} \alpha_r {\mathbf{\,e}}(-ur/R) \beta_s {\mathbf{\,e}}\left(Kr^{-1}/s\right) \right| . \end{align}\]
It remains to observe that for each \(u = 0, \ldots, R-1\) the bound of Lemma 3 applies to the inner sum and implies \[\begin{align} & \left| \mathscr{B}_K(S; {\mathcal{R}}, \boldsymbol{\alpha}, \boldsymbol{\beta}) \right| \\ & \qquad \quad \leqslant AB (RS)^{1/2} \left((R + S)^{1/2} + \left(1 + \frac{K}{RS}\right)^{1/2} \min\{R,S\}\right)N^{o(1)}. \end{align}\] Recalling (15 ), this now simplifies as \[\begin{align} \left| \mathscr{B}_K(S; {\mathcal{R}}, \boldsymbol{\alpha}, \boldsymbol{\beta}) \right| &\leqslant AB (RS)^{1/2} \left(S^{1/2} + \left(1 + \frac{K}{RS}\right)^{1/2}R\right)N^{o(1)}\\ & = AB (RS)^{1/2} \left(S^{1/2} + R+ K^{1/2} S^{-1/2} R^{1/2}\right)N^{o(1)}, \end{align}\] which concludes the proof. ◻
Remark 5. Instead of using Lemma 3, that is, essentially [11], one can also derive a version of Lemma 4 from [11], or from a stronger result due to Bettin and Chandee [14]. However these bounds do not seem to improve our main result.
As we have noticed in Section 2, it is only enough to estimate \(T_{11}(N)\), as we borrow the bounds on \(T_{12}(N)\) and \(T_{2}(N)\) from [1]. Furthermore, we see from (10 ) and (12 ) that it is enough to estimate \(U(N)\) given by (11 ).
We note that it is important to observe that the sum defining \(\psi_H(u)\) in Lemma 2 does not contain the term with \(h=0\), while the sum on the right hand side of (?? ) does. Hence, for any integer \(H\geqslant 1\), by Lemma 2 we have \[\begin{align} U(N) & \ll H^{-1} \sum_{s \geqslant\sqrt{N} } \sum_{\substack{r =1\\\gcd(r,s) = 1}}^{R_s} r \\ & \qquad + \sum_{1\leqslant\left| h \right| \leqslant H} \frac{1}{ h}\left| \sum_{s \geqslant\sqrt{N} } \sum_{\substack{r =1\\\gcd(r,s) = 1}}^{R_s} r {\mathbf{\,e}}\left(hNr^{-1}/s\right) \right|,\\ & \qquad \qquad + \frac{1}{ H} \sum_{1\leqslant\left| h \right| \leqslant H}\left| \sum_{s \geqslant\sqrt{N} }\sum_{\substack{r =1\\\gcd(r,s) = 1}}^{R_s} r {\mathbf{\,e}}\left(hNr^{-1}/s\right) \right|\\ & \ll H^{-1} \sum_{s \geqslant\sqrt{N} } R_s^2 + \sum_{1\leqslant\left| h \right| \leqslant H} \frac{1}{ h}\left| \sum_{s \geqslant\sqrt{N} } \sum_{\substack{r =1\\\gcd(r,s) = 1}}^{R_s} r {\mathbf{\,e}}\left(hNr^{-1}/s\right) \right|\\ & \ll H^{-1} N^{3/2} + \sum_{1\leqslant\left| h \right| \leqslant H} \frac{1}{ h}\left| \sum_{s \geqslant\sqrt{N} } \sum_{\substack{r =1\\\gcd(r,s) = 1}}^{R_s} r {\mathbf{\,e}}\left(hNr^{-1}/s\right) \right|. \end{align}\]
Note that \(R_s\geqslant 1\) implies \(s \ll N\). Therefore, partitioning the corresponding summation over \(s\) into dyadic intervals, we see that there is some integer \(S\) with \[N^{1/2} \ll S \ll N\] and such that \[\label{eq:VUW} U(N) \ll H^{-1} N^{3/2} + V(N,S) \log N,\tag{16}\] where \[V(N,S) = \sum_{1\leqslant\left| h \right| \leqslant H} \frac{1}{ h}\left| \sum_{s \sim S} \sum_{\substack{r =1\\\gcd(r,s) = 1}}^{R_s} r {\mathbf{\,e}}\left(hNr^{-1}/s\right) \right|.\]
Now, if \(S \leqslant H^{1/5} N^{3/5}\) then we use the bound (14 ) and easily derive \[\label{eq:32Small32S} V(N,S) \leqslant N^{1+o(1)} S^{1/2} \leqslant H^{1/10} N^{13/10+ o(1)}.\tag{17}\]
On the other hand, for \(S > H^{1/5} N^{3/5}\), Lemma 4 (used with \(A\ll N/S\) and \(B \ll 1\)), after recalling that \(R \ll N/S\), implies the same bound: \[\begin{align} \sum_{s \sim S} \sum_{\substack{r =1\\\gcd(r,s) = 1}}^{R_s} & r {\mathbf{\,e}}\left(hNr^{-1}/s\right)\\ & \leqslant (N/S) N^{1/2+o(1)} \left(S^{1/2} + NS^{-1} + h^{1/2} N S^{-1} \right)\\ & \leqslant (N/S) N^{1/2+o(1)} \left(S^{1/2} + h^{1/2} N S^{-1} \right). \end{align}\] Therefore, recalling that \(S > H^{1/5} N^{3/5}\), we obtain \[\begin{align} V(N,S) & \leqslant(N/S) N^{1/2+o(1)} \left(S^{1/2} + H^{1/2} N S^{-1} \right)\\ & = N^{3/2+o(1)} S^{-1/2} + H^{1/2} N^{5/2+o(1)} S^{-2}\\ & \leqslant H^{-1/10} N^{6/5+o(1)} + H^{1/10} N^{13/10+o(1)} \\ & \leqslant H^{1/10} N^{13/10+o(1)} . \end{align}\]
Therefore, the bound (17 ) holds for any \(S\). Substituting (17 ) in (16 ) yields \[U(N) \ll H^{-1} N^{3/2} + H^{1/10} N^{13/10+o(1)}\] and choosing \[H = \left\lceil N^{2/11}\right\rceil\] to optimise the bound, we obtain \[U(N) \ll N^{29/22+o(1)}.\]
Finally, recalling (10 ) and (12 ), we derive (6 ) and conclude the proof.
The author is very grateful to Michel Balazard, Bruno Martin and Jens Marklof for encouragement and very stimulating discussions and comments. The author also would like to thank the referee for a careful reading, which revealed some inaccuracies in the original version of the paper.
During the preparation of this work, the author was supported in part by the Australian Research Council Grant DP230100534.