Classification of weights for which uniform weighted convergence implies Cesàro convergence, beyond the tempered regime

Bergelson, Moreira, and Richter proved single and multiple recurrence along functions of an iterated-difference class $\mathcal{F}$ in the sense of uniform Riesz (weighted) means, and conjectured in their Remark 1.12 that for tempered functions these weighted means can be replaced by ordinary Cesàro averages. Reilly confirmed that conjecture, and also proved that the replacement fails in general outside the tempered regime, the function $f(n)=\log n$ supplying a concrete obstruction. The open problem is to classify the weights $W$ (equivalently, the functions $f\in\mathcal{F}$ via $W=\Delta^{\ell-1}f$) for which uniform $W$-weighted convergence of bounded sequences implies Cesàro convergence. Sufficiency is known when $N\cdot\Delta W(N)\to\infty$; failure is known for $W=\log$, where $N\cdot\Delta W(N)\to 1$, and Bergelson–Moreira–Richter's Remark 1.12 cites further faster-than-linear failure examples from the ergodic averaging sequences of Boshernitzan–Kolesnik–Quas–Wierdl. The boundary regimes — $N\cdot\Delta W(N)$ tending to a finite positive constant or to 0, or oscillating — remain unclassified beyond the known failure at $W=\log$.

Background

A probability measure preserving system is a quadruple $(X,\mathscr{B},\mu,T)$ consisting of a probability space together with an invertible, measure-preserving transformation $T$. A central question of ergodic theory is multiple recurrence: for a measurable set $A$ with $\mu(A)>0$, is it true that $\lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^{N}\mu(A\cap T^{-\lfloor f(n)\rfloor}A\cap\cdots\cap T^{-\lfloor f(n+k)\rfloor}A)>0$ when the return times follow a non-polynomial function $f$? By Furstenberg's correspondence principle, such statements are equivalent to combinatorial assertions that every set of integers of positive upper density $\bar d(E)=\limsup_{N\to\infty}|E\cap[1,N]|/N$ contains configurations $\{a,a+\lfloor f(n)\rfloor,\dots,a+\lfloor f(n+k)\rfloor\}$.

Bergelson, Moreira, and Richter (2020) introduced a class $\mathcal{F}$ of functions built by iterated differences: $\mathcal{F}_0$ consists of the functions $f:\mathbb{N}\to\mathbb{R}$ tending to 0 that are eventually monotone with a divergent series, that is $\left|\sum_{n=1}^{\infty}f(n)\right|=\infty$; $\mathcal{F}_{\ell+1}=\{f:\Delta f\in\mathcal{F}_\ell\}$ where $\Delta f(n)=f(n+1)-f(n)$; and $\mathcal{F}=\bigcup_{\ell\geq 1}\mathcal{F}_\ell$. This class contains $n^{\alpha}$ for every non-integer $\alpha>0$ and every tempered function, meaning a function on a half-line that is $\ell$ times continuously differentiable with $f^{(\ell)}$ tending monotonically to 0 and $x|f^{(\ell)}(x)|\to\infty$ (a notion from the theory of uniform distribution). For $f\in\mathcal{F}_\ell$ they set $W=\Delta^{\ell-1}f$ and proved their recurrence and convergence statements (their Theorem D) in the sense of uniform Riesz means with respect to $W$: the $W$-weighted average over an interval is $\mathbb{E}_{n\in[M,N]}^{W}a(n)=\frac{1}{W(N)-W(M)}\sum_{n=M}^{N}\Delta W(n)a(n)$, and the uniform Riesz mean $\mathbb{E}_{\mathrm{unif}}^{W}$ is the limit of these averages over all intervals with $W(N)-W(M)\to\infty$. For $W=\mathrm{Id}$ this recovers the classical uniform Cesàro mean. In Remark 1.12 they conjectured that for tempered $f$ these weighted means can be replaced by ordinary Cesàro averages $\frac{1}{N}\sum_{n=1}^{N}a(n)$, while noting that the replacement provably fails for some functions in $\mathcal{F}$ growing faster than linear, citing examples of Boshernitzan–Kolesnik–Quas–Wierdl.

Reilly (2026) confirmed the conjecture. His Theorem A shows that if $W$ increases to $\infty$, $\Delta W$ eventually decreases to 0, and $N\cdot\Delta W(N)\to\infty$ — the situation attained when $W=\Delta^{\ell-1}f$ for tempered $f$ — then whenever the uniform $W$-weighted averages of a bounded sequence in a Banach space converge to $L$, the Cesàro averages also converge to $L$. His engine (Theorem B) is a chain $(1)\implies(2)\iff(3)\iff(4)\iff(5)$, valid for weights $V:\mathbb{N}\to(0,\infty)$ increasing to $\infty$ with $\Delta\log V(N)$ eventually decreasing to 0 and $\Delta\log V(N)/\Delta\log N\to\infty$, linking: (1) ordinary $V$-weighted averages $\lim_{N\to\infty}\mathbb{E}_{n\leq N}^{V}x_n$; (2) iterated $V$-weighted averages $\mathbb{E}(\infty)^{V}x_n$, obtained by iterating the initial-segment averaging operator $\mathbb{E}_{n\leq N}^{V}$ and letting the number of iterations tend to infinity; (3) $U$-weighted averages for auxiliary weights $U$ increasing to $\infty$ that satisfy the admissibility condition (his equation (1.15))

$\lim_{N\to\infty}\frac{V(N-1)}{\Delta U(N)}\left(\frac{\Delta U(N)}{\Delta V(N)}-\frac{\Delta U(N-1)}{\Delta V(N-1)}\right)=-1$

written in Reilly's backward-difference convention $\Delta f(n)=f(n)-f(n-1)$; (4) Cesàro averages over short terminal intervals $[N-s(N),N]$ with $s(N)\cdot\Delta\log V(N)\to\infty$; and (5) uniform $\log V$-weighted averages. When $U$ and $V$ belong to the same Hardy field — a field of real functions on a half-line, closed under differentiation modulo eventual equality, whose members are eventually monotone — condition (1.15) reduces to the checkable requirement $\log U(N)/\log V(N)\to 0$ (Remark 1.9 and Theorem C).

On the negative side, a classical theorem of Schatte (1990) identifies uniform logarithmic limitability — convergence of the uniform averages with weight $W=\log$ — with the limit of iterated Cesàro means, and Reilly's Example 5.2 exhibits the bounded sequence $x_n=e^{2\pi i\log n}$, whose uniform logarithmic averages converge to 0 while its Cesàro averages fail to converge; here $W=\log$ lies in $\mathcal{F}_1$ but $N\cdot\Delta W(N)\to 1$, so the replacement fails precisely at the boundary of the tempered regime. Bergelson–Moreira–Richter's Remark 1.12 additionally cites, from the ergodic averaging sequences studied by Boshernitzan, Kolesnik, Quas, and Wierdl — sequences $(a_n)$ for which the averages $\frac{1}{N}\sum_{n=1}^{N}g\circ T^{a_n}$ converge for every system — further failure examples along faster-than-linear functions (their Theorem C concerns the sequences $\lfloor n^{k}+\log n\rfloor$, logarithmic perturbations of polynomials). What no existing work determines is the exact boundary between the regime $N\cdot\Delta W(N)\to\infty$, where the implication holds, and these known failures.

Problem Statement

Determine a complete classification of the functions $f\in\mathcal{F}$ — equivalently, of the weights $W=\Delta^{\ell-1}f$ for $f\in\mathcal{F}_\ell$, which increase to $\infty$ with $\Delta W$ eventually decreasing to 0 — for which the following implication holds: whenever $(x_n)$ is a bounded sequence in a Banach space and the uniform $W$-weighted averages $\mathbb{E}_{\mathrm{unif}}^{W}(x_n)$ converge to $L$, the Cesàro averages $\frac{1}{N}\sum_{n=1}^{N}x_n$ also converge to $L$. In ergodic terms, this classifies the functions $f\in\mathcal{F}$ for which the uniform Riesz means in Theorem D of Bergelson–Moreira–Richter can be replaced by ordinary Cesàro averages.

Reilly's Theorem A establishes sufficiency of the condition $N\cdot\Delta W(N)\to\infty$ (with $W$ increasing to $\infty$ and $\Delta W$ eventually decreasing to 0), and the known counterexamples show failure in specific regimes: $W=\log$, where $N\cdot\Delta W(N)\to 1$ (Reilly's Example 5.2, via Schatte's theorem), and the faster-than-linear examples cited by Bergelson–Moreira–Richter's Remark 1.12 from Boshernitzan–Kolesnik–Quas–Wierdl's theory of ergodic averaging sequences. The open boundary includes: weights with $N\cdot\Delta W(N)$ tending to a finite positive constant or to 0 (for instance $W(N)=(\log N)^{\alpha}$ with $0<\alpha<1$), and weights for which $N\cdot\Delta W(N)$ oscillates.

References

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