arXiv:2609.16554v1

Weighted averages and applications to sets of multiple recurrence

math.DSmath.CO40C9937A4411K36

Paper metadata

Original submission: September 15, 2026
Claimed MSC: 40C99, 37A44, 11K36
arXiv categories: math.DS, math.CO
Comments: 58 pages

Abstract

We introduce new techniques for determining combinatorial properties of sets of multiple recurrence by considering weighted averages with quickly growing weights. Our main result is a far-reaching generalization of Szemerédi's Theorem which additionally confirms a conjecture of Bergelson-Moreira-Richter and contains as special cases both the Polynomial Szemerédi Theorem due to Bergelson-Leibman-Lesigne and the fact that if $f$ belongs to a broad class of smooth functions and satisfies $x^{d-1}\prec f(x)\prec x^d$ for some $d\in \mathbb{N}$ then for any $\ell\in \mathbb{N}$ , any invertible measure preserving system $(X,\mathscr{B},μ,T)$ , and any $A\in \mathscr{B}$ with $μ(A)>0$ , the set $\{n\in \mathbb{N}: μ(A\cap T^{-[f(n)]}A\cap T^{-2[f(n)]}A\cap \cdots\cap T^{-\ell[f(n)]}A )>0\}$ is thick, meaning that it contains arbitrarily long intervals of natural numbers. Additionally, we formulate and prove a generalization to weighted averages of Boshernitzan's criterion for uniform distribution which we use in the proof of our main result.

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Audit summary

Audited against arXiv v1

Not a correctness certificate. A green “no issues found” result may include yellow suspected typos or minor formal issues that do not affect substantive soundness. It means this audit found no unresolved substantive error under the stated criteria; it does not replace expert scrutiny or formal verification.

Current report

Detailed mathematical audit

Generated October 24, 2026

01Statements3 reported findingsSuspected incorrect statements

The paper develops a substantial weighted-averages framework, but two stated combinatorial claims are false as written, and central recurrence and weighted-uniform-distribution conclusions are not fully supported by the supplied proofs. The defects are localized and do not amount to counterexamples to the main ergodic theorem itself.

Theorem DSuspected incorrect statement

The correspondence principle gives lower bounds, not the asserted equalities

PDF pp. 14–15 · equations (2.16)–(2.19) · arXiv:2609.16554v1

Theorem 2.6 supplies only $\overline d(E\cap(E-n_1)\cap\cdots\cap(E-n_\ell))\geq\mu(A\cap T^{-n_1}A\cap\cdots\cap T^{-n_\ell}A)$ . Averaging this inequality yields a uniform weighted lower-limit bound; it does not give existence of the average of the upper densities, much less equality in (2.16). The equality is false even for $\ell=1$ , $f(x)=x^{3/2}$ , and $W(x)=x$ . Fix $0<\delta<1$ and partition $\mathbb N$ into consecutive blocks of lengths $1,2,3,\ldots$ , putting the first $\lfloor\delta j\rfloor$ points of block $j$ into $E$ . Then $d(E)=\delta$ , the block-boundary set has density zero, and $d(E\cap(E-k))=\delta$ for every fixed integer $k$ . Thus the left side of (2.16) is $\delta$ , whereas the claimed value is $\delta^2$ . The valid replacements are lower-limit inequalities $\liminf E^W_{\mathrm{unif}}\,\overline d(\cdots)\geq\overline d(E)^{\ell+1}$ in part (i) and $>0$ in part (ii); these still imply the advertised $W$ -syndeticity consequences. The printed exponent $k+1$ should also be $\ell+1$ .

Lemma 3.13Suspected incorrect statement

The claimed simultaneous approximation and interval scale are false

PDF pp. 22–24 · Lemma 3.13 and equations (3.22)–(3.25) · arXiv:2609.16554v1

The first assertion includes $i=d$ and therefore requires $\Delta^d f(N)\bmod 1<\varepsilon\Delta^d f(N)$ ; for large $N$ , $0<\Delta^d f(N)<1$ , so this is impossible when $0<\varepsilon<1$ . Removing that endpoint does not fix the proof: the induction obtains upper bounds involving $(s(N_1)\Delta^d f(N_1))^{i/d}$ , while the displayed conclusion drops the factor $s(N_1)^{i/d}$ in the wrong direction. The additional interval-length assertion is also false. For $d=2$ and $f(x)=x^{3/2}$ , one has $\Delta^2f(N)\sim(3/4)N^{-1/2}$ and the lemma's $W(N)=(\Delta^2f(N))^{-1/2}\asymp N^{1/4}$ . If $f(N+j)\bmod1$ stays in a fixed interval $(0,\varepsilon)$ with $\varepsilon<1/8$ , lifting those fractional parts to $(0,\varepsilon)$ gives a bounded discrete-convex sequence with second differences $\Delta^2f(N+j)$ . The elementary convexity bound forces $K=O(N^{1/4})$ , contradicting the claimed $K/W(N)\to\infty$ .

Theorems 1.4, E, and ANot independently verified

The principal recurrence and convergence conclusions remain unverified

PDF pp. 3–7, 12–13, 15–32, and 47–55 · arXiv:2609.16554v1

The proof of Theorem 1.4 uses the false final assertion of Lemma 3.13 to declare a fractional-part set $W$ -thick. The positive implication of Theorem E relies on Theorem 3.19, whose short-interval induction does not cover the stated hypotheses; this implication is then used in the horizontal-character verification for Theorem 6.1 and hence in the proof of Theorem A. No counterexample to Theorems 1.4, E, or A is supplied, and the general PET/nilsystem reduction is consistent after the local notation corrections listed under Proofs, but the manuscript does not close these two analytic inputs.

02Proofs4 reported findingsSuspected incomplete or incorrect proofs

The correspondence and Lemma 3.13 arguments contain explicit false steps, and the short-interval exponential-sum proof misses an entire parameter regime. Several later PET and horizontal-character defects are mechanically repairable notation errors.

Proof of Lemma 3.13Suspected incorrect as written · no repair proposed

The final bounds do not follow from the constructed intervals

PDF pp. 22–24 · equations (3.23)–(3.25) and final paragraph · arXiv:2609.16554v1

With $A=s\Delta^df$ and $B_i=\varepsilon(s\Delta^df)^{1-i/d}$ , the induction gives $K_d\gtrsim(s\Delta^df)^{-1/d}$ and bounds the lower differences by powers of $s\Delta^df$ . Since $s\to\infty$ , these bounds are weaker, not stronger, than the powers of $\Delta^df$ asserted in the statement, and $K_d=o((\Delta^df)^{-1/d})$ rather than $K_d/(\Delta^df)^{-1/d}\to\infty$ . Lemma 3.12 only permits replacing $\Delta^df(N_1)$ by $\Delta^df(N)$ ; it cannot remove the factor $s$ . The false thickness conclusion is used on p. 13 to intersect the recurrence set with the required rounding-correction set, so the proof of Theorem 1.4 does not close.

Proof of Theorem 3.19Suspected incomplete as written · plausible repair, not fully checked

The derivative-test case split leaves a concrete admissible regime untreated

PDF pp. 31–32 · proof of Theorem 3.19 · arXiv:2609.16554v1

The theorem allows an arbitrary nondecreasing $s:\mathbb N\to\mathbb N$ , but the proof treats $s(x)$ as a Hardy function and assumes that $s(x)^d f^{(d+1)}(x)$ has either an infinite or a finite limit. More decisively, in the second case it sets $j=d+1$ and asserts $X^j\lambda=s(N)^{d+1}f^{(d+1)}(N)\to\infty$ from the hypothesis $s(N)^df^{(d)}(N)\to\infty$ . This inference fails: for $d=2$ , $f(x)=x^{3/2}$ , and $s(N)=\lfloor N^{3/10}\rfloor$ , the theorem's hypothesis holds, the proof enters its second case, but $s(N)^3|f^{(3)}(N)|\to0$ . A likely route is to split according to $s(N)^{d-1}f^{(d)}(N)$ and apply the $d$ th-derivative estimate on the complementary indices, together with a subsequence argument for arbitrary $s$ ; that argument is not present.

Derivation of Theorem DSuspected incorrect as written · verified weaker conclusion

A one-sided correspondence inequality is silently promoted to convergence and equality

PDF pp. 14–15 · paragraph following Theorem 2.6 · arXiv:2609.16554v1

There is no intervening argument after the correspondence principle that could reverse its inequality or prove convergence of the upper-density sequence. Keeping the displayed inequality and taking uniform weighted lower limits gives exactly the corrected lower bounds described under Statements. Since every correlation lies in $[0,1]$ , these lower bounds also give the claimed $W$ -syndetic level sets. Thus the applications survive after weakening (2.16) and (2.18), but the printed equalities do not.

Theorem 5.2 and the proof of Theorem 6.1Suspected typo · an apparently sound repair

Two later reductions contain mechanically identifiable notation defects

PDF p. 41 · proof of Theorem 5.2; pp. 55–56 · equations (6.22) and the following display · arXiv:2609.16554v1

On p. 41 the PET induction defines $F_i=[f_i-f_1]$ and $F_{\ell+i}=[f_i(\cdot+h)-f_1]$ and then treats the $F_i$ as Hardy functions; the uniquely intended definitions omit these inner rounding brackets. The same paragraph's $g_{k+i}$ , $\{F_1,\ldots,F_{2k}\}$ , $F_{2k}$ , and $f_k$ must use $\ell$ in place of the undefined $k$ . On pp. 55–56, (6.22) and the concluding display omit $2\pi i$ from the exponential and omit the weighted average from the limit; restoring them gives $\lim_N E^W_{n\leq N}\eta(w_r(n)\Gamma)=\rho\lim_N E^W_{n\leq N}e^{2\pi i f(qn+r)}=0$ . The needed unboundedness of $f$ follows directly because nontriviality of $\eta$ on $z_1^{\mathbb R}\cdots z_m^{\mathbb R}\Gamma$ makes at least one coefficient of the strictly ordered $g_j$ expansion nonzero. These repairs preserve the intended PET and horizontal-character reductions.

03Novelty0 reported findingsNo novelty concerns found

No non-novelty findings.

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Paper: arXiv:2609.16554v1
Authors listed: Vitaly Bergelson, Michael Reilly
Audit date: October 24, 2026

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Weighted averages and applications to sets of multiple recurrence

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

The correspondence principle gives lower bounds, not the asserted equalities

The claimed simultaneous approximation and interval scale are false

The principal recurrence and convergence conclusions remain unverified

The final bounds do not follow from the constructed intervals

The derivative-test case split leaves a concrete admissible regime untreated

A one-sided correspondence inequality is silently promoted to convergence and equality

Two later reductions contain mechanically identifiable notation defects

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