June 14, 2026
The aim of this note is to illustrate that the set of integrable functions on the torus \(\mathbb{T}\) for which the Fourier series diverges almost everywhere is of the second Baire category in \(L^1(\mathbb{T})\).
In 1923, Kolmogorov [1], as a nineteen-year-old student, constructed an integrable function on the torus \(\mathbb{T}:=\mathbb{R}/\mathbb{Z}\) whose Fourier series diverges almost everywhere. Three years later, Kolmogorov [2] improved on this by creating an integrable function on \(\mathbb{T}\) whose Fourier series diverges everywhere on \(\mathbb{T}\). Kolmogorov’s seminal papers [1], [2] initiated new directions of research in Fourier series and engendered an extensive literature on diverging Fourier series. An important simplification and several refinements of Kolmogorov’s examples appear in Stein’s article [3], which also covers the case of the Walsh–Paley series. For a comprehensive historical background and a thorough bibliography on the subject, we refer to Ul’yanov’s survey [4] as well as to the books of Zygmund [5] and Grafakos [6]. In particular, Grafakos’ books [6], [7] offer a modern treatment to the subject.
Kolmogorov’s results arose from his attempt to disprove Luzin’s conjecture from 1913, which asserts that the partial sums of the Fourier series of a square-integrable function on \(\mathbb{T}\) converge almost everywhere. At that time, there was a strong belief that Luzin’s conjecture was likely false.
However, four decades after Kolmogorov’s papers, Carleson [8] proved Luzin’s conjecture in the affirmative for \(L^2(\mathbb{T})\) functions and Hunt [9] subsequently extended this result to all \(L^p(\mathbb{T})\) functions with \(p \in (1, \infty)\). We refer to Fefferman’s paper [10] and to Lacey and Thiele’s paper [11] for different proofs of Carleson’s theorem, as well as to Grafakos [7], where the Carleson–Hunt theorem is presented in great detail for \(L^p(\mathbb{T})\) functions with \(p \in (1, \infty)\).
Collectively, Kolmogorov’s papers [1], [2] and the Carleson–Hunt theorem [8], [9] reveal that Kolmogorov’s examples are obstructions which characterize \(L^1(\mathbb{T})\)-space phenomena for Fourier series. This advances a natural question: how large topologically (in the sense of Baire category) is the set of all integrable functions \(f \in L^1(\mathbb{T})\) whose Fourier series diverges almost everywhere on \(\mathbb{T}\)?
In this short note, we give an ad-hoc argument based on Kolmogorov’s methods [1], [2], (see Proposition 1 below), illustrating that the set of integrable functions whose Fourier series diverge almost everywhere on \(\mathbb{T}\) is topologically generic in \(L^1(\mathbb{T})\). In other words, this property holds on a residual set, i.e., a countable intersection of dense open sets. Loosely speaking, Kolmogorov’s pathological examples are topologically quite common in \(L^1(\mathbb{T})\).
We thank Leonidas Daskalakis, Jacek Dziubański, Dariusz Kosz and Jim Wright for helpful comments on an earlier draft of this paper.
Throughout the paper the set of positive integers and the set of nonnegative integers will be denoted, respectively, by \(\mathbb{Z}_+ \mathrel{\vcenter{:}}= \{1, 2, \ldots\}\) and \(\mathbb{N} \mathrel{\vcenter{:}}= \{0,1,2,\ldots\}\). The sets \(\mathbb{Z}\), \(\mathbb{R}\), and \(\mathbb{C}\) have their standard meaning. We also denote \(\mathbb{R}_+ \mathrel{\vcenter{:}}= (0, \infty)\). The torus \(\mathbb{T} := \mathbb{R}/\mathbb{Z}\) is endowed with the Lebesgue measure \(d\mu_{\mathbb{T}}(x):=dx\) inherited from the real line \(\mathbb{R}\), and \(\mu_{\mathbb{T}}(E):=\int_Edx\) denotes the Lebesgue measure of a Lebesgue measurable set \(E \subseteq \mathbb{T}\).
All vector spaces here will be defined over the complex numbers \(\mathbb{C}\). The space of all measurable functions \(f:\mathbb{T}\to \mathbb{C}\) whose modulus is integrable with \(p\)-th power is denoted by \(L^p(\mathbb{T})\) for \(p\in[1, \infty)\), whereas \(L^{\infty}(\mathbb{T})\) denotes the space of all essentially bounded measurable functions on \(\mathbb{T}\).
Let \(e(x):=e^{2\pi i x}\) for \(x\in \mathbb{R}\). For \(f\in L^1(\mathbb{T})\) the Fourier coefficients are defined by \[\widehat{f}(n):=\int_{\mathbb{T}}f(x)e(-x n)dx \quad \text{ for } \quad n\in\mathbb{Z}.\] The partial sums of the Fourier series corresponding to \(f\in L^1(\mathbb{T})\) are defined by \[S_N(f)(x):=\sum_{n\in[-N, N]\cap \mathbb{Z}}\widehat{f}(n)e(x n)\quad \text{ for } \quad x\in\mathbb{T}, \;N\in\mathbb{N}.\] Using the Dirichlet kernel \(D_N(x):=\sum_{n\in[-N, N]\cap \mathbb{Z}}e(x n)\), we see that \[S_N(f)(x)=D_N*f(x)=\int_{\mathbb{T}}f(x-y)D_N(y)dy.\]
The main theorem of the paper reads as follows.
Theorem 1. The set \[\begin{align} \label{eq:1} \mathcal{D}:=\big\{f\in L^1(\mathbb{T}): \mu_{\mathbb{T}}(\{x\in\mathbb{T}: \sup_{N\in\mathbb{N}}|D_N*f(x)|=\infty\})=1\big\} \end{align}\tag{1}\] is of the second Baire category in \(L^1(\mathbb{T})\). In particular, the set of all integrable functions whose Fourier series diverges almost everywhere on \(\mathbb{T}\) is of the second Baire category in \(L^1(\mathbb{T})\).
From Theorem 1 it follows immediately that there exist integrable functions whose Fourier series diverge almost everywhere, in fact with unbounded partial sums; however, our proof should not be regarded as an alternative method for producing Kolmogorov’s examples [1], [2], but rather as a complement to his approach, providing a topological characterization of this phenomenon.
Interestingly, Marcinkiewicz [12] constructed an integrable function on \(\mathbb{T}\) whose Fourier series diverges almost everywhere, yet whose partial sums are bounded at every point of \(\mathbb{T}\).
We will prove Theorem 1 by showing that \(\mathcal{D}^c\) is of the first category in the sense of Baire category, in other words it can be written as a countable union of nowhere dense sets in \(L^1(\mathbb{T})\).
A key tool will be the following result of Kolmogorov [1], [2].
Proposition 1. For each \(M\in\mathbb{R}_+\) there exists a trigonometric polynomial \(g_M\) and a measurable set \(G_M \subseteq \mathbb{T}\) with measure \(\mu_{\mathbb{T}}(G_M) > 1 - 2^{-M}\) such that \(\|g_M\|_{L^1(\mathbb{T})}=1\), and satisfying \[\begin{align} \label{eq:5} \inf_{x\in G_M}\sup_{k\in\mathbb{Z}_+}|D_k*g_M(x)|>2^M. \end{align}\qquad{(1)}\]
Proof. For a detailed proof we refer to [6]. ◻
Proposition 1 (see also [5]) was a key mechanism behind the construction of Kolmogorov’s [1], [2] explicit examples; it will also be essential in our categorical argument.
For \(m, n\in\mathbb{Z}_+\), define \[\begin{align} \label{eq:4} F(m, n):=\Big\{f\in L^1(\mathbb{T}): \mu_{\mathbb{T}}(\{x\in\mathbb{T}: \sup_{N\in\mathbb{N}}|D_N*f(x)|>n\})\le \frac{m}{m+1}\Big\}. \end{align}\tag{2}\]
Lemma 1. The sets \(F(m, n)\) defined in 2 are closed in \(L^1(\mathbb{T})\) for every \(m, n\in\mathbb{Z}_+\).
Proof. Fix \(m, n\in\mathbb{Z}_+\), equivalently it suffices to show that \(F(m, n)^c\) is open in \(L^1(\mathbb{T})\). Pick \(f\in F(m, n)^c\) and we show that there exists \(\eta>0\) such that for any \(g\in L^1(\mathbb{T})\) when \(\|f-g\|_{L^1(\mathbb{T})} < \eta\), then \(g\in F(m, n)^c\). Define two sets \[\begin{align} E_N(f)&:=\big\{x \in\mathbb{T}: \sup_{1 \leq k \leq N}|D_k * f(x)|>n\big\},\\ E_\infty(f)&:=\big\{x \in\mathbb{T}: \sup_{k\in\mathbb{N}}|D_k * f(x)|>n\big\}. \end{align}\] Observe that \(E_N\) is an increasing sequence of sets, that is \(E_{N} \subseteq E_{N+1}\). Using continuity from below of the measure, we have \(\lim_{N\to \infty}\mu_{\mathbb{T}}(E_N(f)) = \mu_{\mathbb{T}}(E_\infty(f))\). Since \(f\in F(m, n)^c\), it follows that there exists some \(N \in \mathbb{N}\) such that \(\mu_{\mathbb{T}}(E_N(f)) > \frac{m}{m+1}\). Define another set \[\begin{align} E_{N, \delta}(f):=\big\{x \in\mathbb{T}: \sup_{1 \leq k \leq N}|D_k * f(x)| > n+\delta\big\}. \end{align}\] We note that \(E_{N, \delta_2}(f)\subseteq E_{N, \delta_1}(f)\) for \(\delta_2>\delta_1>0\). Hence, we can choose \(\delta>0\) such that \(\mu_{\mathbb{T}}(E_{N, \delta}(f)) > \frac{m}{m+1}+\delta\). Set \(\eta:= \min_{1 \leq k \leq N} \delta^22^{-k}\|D_k\|_{L^1(\mathbb{T})}^{-1}\) and observe that for any \(g\in L^1(\mathbb{T})\) satisfying \(\|f-g\|_{L^1(\mathbb{T})} < \eta\), we obtain, for any \(k\in\{1, \ldots, N\}\), by Chebyshev’s inequality followed by Young’s convolution inequality that \[\begin{align} \mu_{\mathbb{T}}\big(\big\{x \in\mathbb{T}: |D_k *(f-g)(x)| \geq \delta\big\}\big)\le \frac{1}{\delta}\|D_k\|_{L^1(\mathbb{T})}\|f-g\|_{L^1(\mathbb{T})}<\frac{\delta}{2^k}. \end{align}\] Taking \[\begin{align} B:=\bigcup_{k=1}^N\{x\in \mathbb{T}:\left|D_k *(f-g)(x)\right| \geq \delta\}, \end{align}\] we see that \(\mu_{\mathbb{T}}(B)\le \sum_{k=1}^N\frac{\delta}{2^k}<\delta\). Moreover, we have \[\begin{align} E_{N, \delta}(f)\subseteq \big\{x \in\mathbb{T}: \sup_{1 \leq k \leq N}|D_k * g(x)|>n\big\}\cup B. \end{align}\] Therefore, \(\mu_{\mathbb{T}}(E_{N}(g))\ge \mu_{\mathbb{T}}(E_{N, \delta}(f))-\mu_{\mathbb{T}}(B)> \frac{m}{m+1}+\delta-\delta=\frac{m}{m+1}\), which implies that \(g\in F(m, n)^c\). This in turn yields that \(F(m, n)^c\) is open and the proof of the lemma follows. ◻
Lemma 2. The sets \(F(m, n)\) defined in 2 have empty interiors for every \(m, n\in\mathbb{Z}_+\).
Proof. We fix \(m, n\in\mathbb{Z}_+\) and we will show that \(F(m, n)\) has empty interior. Equivalently, we prove that for all \(f \in F(m, n)\) and for all \(\varepsilon>0\), there exists \(h \in L^1(\mathbb{T})\) such that \(\|f-h\|_{L^1(\mathbb{T})} < \varepsilon\) and \(h \not \in F(m, n)\). By a simple density argument, there exists a trigonometric polynomial \(g\in L^1(\mathbb{T})\) such that \(\|f-g\|_{L^1(\mathbb{T})} < \varepsilon/2\). Then there exists a finite constant \(K\in\mathbb{R}_+\) such that \[\begin{align} \sup_{x\in\mathbb{T}}\sup _{N \in \mathbb{Z}_+}|D_N * g(x)| \leq K. \end{align}\] Choose \(M\in\mathbb{Z}_+\) so that \[\begin{align} 2^{-M/2} < \frac{\varepsilon}{2}, \quad \text{ and } \quad 2^{M/2} > n+K, \quad \text{ and } \quad 1-2^{-M}>\frac{m}{m+1}. \end{align}\] Define the perturbed function \(h:= g + 2^{-M/2}g_M\), where \(g_M\) is a polynomial as in Proposition 1. By our choice of \(M\in\mathbb{Z}_+\) we see that \(\|f-h\|_{L^1(\mathbb{T})} < \varepsilon\), since \(\|g_M\|_{L^1(\mathbb{T})}=1\) and \[\begin{align} \|f-h\|_{L^1(\mathbb{T})} \leq \|f-g\|_{L^1(\mathbb{T})} + \|g-h\|_{L^1(\mathbb{T})} = \|f-g\|_{L^1(\mathbb{T})}+ 2^{-M/2} \|g_M\|_{L^1(\mathbb{T})} < \frac{\varepsilon}{2}+ 2^{-M/2}. \end{align}\] It suffices to show that \(h \not \in F(m, n)\). Indeed, if \(x \in G_M\), then by the triangle inequality, we have \[\begin{align} \sup_{k\in\mathbb{Z}_+}|D_k * h(x)|&=\sup_{k\in\mathbb{Z}_+}|D_k * (g+2^{-M / 2} g_M)(x)| \\ &\geq 2^{-M / 2} \sup_{k\in\mathbb{Z}_+}|D_k * g_M(x)|-\sup_{k\in\mathbb{Z}_+}|D_k * g(x)| \\ & > 2^{-M/2} 2^M - K = 2^{M/2} -K. \end{align}\] By our choice of \(M\in\mathbb{Z}_+\), we conclude that \[\begin{align} \sup_{k\in\mathbb{Z}_+}|D_k * h(x)| > n, \end{align}\] which also guarantees that \(G_M \subseteq \{x\in\mathbb{T} : \sup_{k\in\mathbb{Z}_+}|D_k * h)(x)| >n\}\), and yields \[\begin{align} \mu_{\mathbb{T}}\big(\big\{x \in\mathbb{T}: \sup_{k\in\mathbb{Z}_+}|D_k * h(x)|>n\big\}\big)\ge \mu_{\mathbb{T}}(G_M)>1-2^{-M}>\frac{m}{m+1}. \end{align}\] Therefore, \(h \not \in F(m, n)\) and consequently \(F(m, n)\) has empty interior as desired. ◻
Proof of Theorem 1. Observe that \[\begin{align} \mathcal{D}^c= \big\{f\in L^1(\mathbb{T}): \mu_{\mathbb{T}}(\{x\in\mathbb{T}: \sup_{N\in\mathbb{N}}|D_N*f(x)|=\infty\})<1\big\}. \end{align}\] For \(f\in L^1(\mathbb{T})\), define \[\begin{align} A_n:=\{x \in\mathbb{T}: \sup _{N \in \mathbb{N}}|D_N * f(x)|>n\} \quad \text{ for } \quad n\in\mathbb{Z}_+. \end{align}\] Observe that \(A_n\) is a decreasing sequence of sets, that is \(A_{n+1} \subseteq A_n\). Further \(\mu_{\mathbb{T}}(A_1) \leq 1\). So, we may invoke continuity from above of the measure to see that \[\begin{align} \mu_{\mathbb{T}}(\{x\in\mathbb{T}: \sup_{N\in\mathbb{N}}|D_N*f(x)|=\infty\})=\lim_{n\to \infty}\mu_{\mathbb{T}}(A_n)\le 1. \end{align}\] Hence, we obtain \[\begin{align} \mathcal{D}^c\subseteq \bigcup_{n\in\mathbb{Z}_+} \big\{f\in L^1(\mathbb{T}): \mu_{\mathbb{T}}(\{x\in\mathbb{T}: \sup_{N\in\mathbb{N}}|D_N*f(x)|>n\})<1\big\}. \end{align}\] Consequently, using the sets \(F(m, n)\) from 2 , we can further write \[\begin{align} \label{eq:3} \mathcal{D}^c\subseteq \bigcup_{n\in\mathbb{Z}_+}\bigcup_{m\in\mathbb{Z}_+} F(m, n). \end{align}\tag{3}\] The set \(\bigcup_{n\in\mathbb{Z}_+}\bigcup_{m\in\mathbb{Z}_+} F(m, n)\) is of the first category, since each \(F(m, n)\) is closed and has empty interior by Lemma 1 and Lemma 2. This ensures that \(\mathcal{D}^c\) is of the first category as a subset of a first category set in 3 . The proof of Theorem 1 now follows. ◻
The authors were partially supported by the NSF CAREER grant (DMS-2236493).↩︎