arxiv: 2604.18535 · v2 · submitted 2026-04-20 · 🧮 math.CA · math.PR

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Counterexamples for lacunary dilates via dyadic spike blocks

Boon Suan Ho

Pith reviewed 2026-05-10 03:23 UTC · model grok-4.3

classification 🧮 math.CA math.PR

keywords dyadic spike blockslacunary dilatesErdős problemspointwise divergencemean-zero functionsL^q spacesFourier averagescounterexamples

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The pith

Dyadic spike blocks produce mean-zero L^q functions where lacunary dilate averages diverge almost everywhere at the endpoint.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper builds counterexamples to two open questions of Erdős on the pointwise behavior of dilates of functions on the circle. It introduces dyadic spike blocks consisting of rare positive spikes that generate long positive runs in the averages, paired with a fixed lower floor that blocks cancellation from later stages. With a careful choice of parameters, this yields a mean-zero function belonging to every L^q space for finite q, together with a lacunary sequence of dilates whose ratios are at least 2, such that the function is well approximated in L^2 by its partial sums yet the averages diverge to positive infinity almost everywhere. The same method also supplies, for every p at least 2, an L^p function whose averages diverge faster than the expected rate. These constructions show that positive results known for exponents strictly larger than 1/2 fail at the endpoint 1/2 and give negative answers to the two Erdős problems.

Core claim

The endpoint construction gives a mean-zero f in the intersection over 1 ≤ q < ∞ of L^q(T) and a sequence n_j = 2^{m_j} with n_{j+1}/n_j ≥ 2 such that ||f − S_N f||_2 ≪ (log log N)^{-1/2} while limsup (1/N) ∑_{j≤N} f(n_j x) = +∞ for almost every x. A second parameter choice produces, for every 2 ≤ p < ∞, an f in L^p(T) satisfying limsup [∑_{j≤N} f(n_j x)] / [N (log N)^{1/p − ε}] = +∞ almost everywhere for every ε > 0. The case p = 2 answers Erdős Problem #995, and the endpoint construction answers Erdős Problem #996 negatively. A bounded small-set companion construction is also given.

What carries the argument

The dyadic spike block, which places rare positive spikes to create long positive runs in the lacunary averages while a deterministic lower floor prevents cancellation from the remaining stages.

Load-bearing premise

The dyadic spike blocks can be arranged so that their positive runs survive without being canceled by contributions from later stages.

What would settle it

An explicit construction or numerical check in which the attempted spike blocks produce averages whose limsup remains finite almost everywhere or in which the L^2 approximation error fails to stay below (log log N)^{-1/2}.

Figures

Figures reproduced from arXiv: 2604.18535 by Boon Suan Ho.

**Figure 1.** Figure 1: The spike ϕd. Here hd := √ 2 d − 1 and gd := 1/ √ 2 d − 1. Lemma 2.1 (Distribution, independence, and Fourier support). Let d ≥ 1. (a) The random variable ϕd(x) takes the values hd and −gd with probabilities 2 −d and 1 − 2 −d respectively. (b) If 0 ≤ v1 < · · · < vs and vi+1 − vi ≥ d, then the random variables ϕd(2vix), 1 ≤ i ≤ s, are independent. (c) For r ̸= 0, ϕcd(r) = 0 whenever 2 d | r. Hence, for v ≥… view at source ↗

read the original abstract

We construct dyadic lacunary counterexamples for two problems of Erd\H{o}s on pointwise behavior of dilates on the circle. The main device is a dyadic spike block: rare positive spikes create long positive runs in the lacunary averages, while a deterministic lower floor prevents cancellation from the remaining stages. The endpoint construction gives a mean-zero $f\in\bigcap_{1\le q<\infty}L^q(\mathbb T)$ and a sequence $n_j=2^{m_j}$, $n_{j+1}/n_j\ge2$, such that $$ \|f-S_Nf\|_2\ll (\log\log N)^{-1/2}, \qquad \limsup_{N\to\infty} \frac1N\sum_{j\le N}f(n_jx)=+\infty $$ for almost every $x$. Thus Matsuyama's positive theorem at exponent $c>1/2$ cannot be extended to the endpoint $c=1/2$, and Erd\H{o}s Problem #996 has a negative answer. A second choice of parameters gives, for every $2\le p<\infty$, functions $f\in L^p(\mathbb T)$ with $$ \limsup_{N\to\infty} \frac{\sum_{j\le N}f(n_jx)} {N(\log N)^{1/p-\varepsilon}} =+\infty \qquad(\varepsilon>0) $$ almost everywhere; the case $p=2$ answers Erd\H{o}s Problem #995. We also include a bounded small-set companion construction.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives explicit counterexamples to Erdős problems 995 and 996 by building lacunary divergence via dyadic spike blocks with a lower floor to control cancellation.

read the letter

The core contribution is a direct construction that answers two open questions negatively. For the endpoint case it produces a mean-zero f in all L^q (q finite) along with lacunary n_j = 2^{m_j} (ratio at least 2) such that the L2 distance to the partial sums decays like (log log N)^{-1/2} while the averages (1/N) sum f(n_j x) have limsup +infty almost everywhere. A second parameter choice yields the corresponding divergence rate for every L^p, p >= 2, which covers problem 995 at p=2. The device is the dyadic spike block: rare tall positive spikes create long positive runs in the averages, and a deterministic lower floor on dyadic intervals is added to keep other stages from canceling them out under the lacunary dilations. This seems to be the first time this style of block has been used for these particular problems, and the parameter tuning looks deliberate enough to hit the claimed rates without blowing up the integrability. The main soft spot is whether the lower floor really guarantees the limsup on a set of full measure. The lacunarity helps separate scales, but the measure of the sets where negative spikes from non-aligned blocks could still win has to be controlled tightly; if that estimate slips even a little, the a.e. statement could fail while the L2 rate and L^q membership still hold. The paper claims the parameters are chosen so the floor dominates, but those measure calculations are the part that needs the closest check. This is squarely for people working on pointwise convergence of lacunary series and related ergodic questions. The work is a clean negative result built from explicit functions rather than abstract existence, so it deserves a serious referee to verify the estimates. I would send it to review.

Referee Report

2 major / 3 minor

Summary. The manuscript constructs counterexamples for two Erdős problems on the pointwise behavior of lacunary dilates on the circle. The central device is the dyadic spike block, which uses rare positive spikes to produce long positive runs in the averages while a deterministic lower floor prevents cancellation from other stages. The endpoint construction yields a mean-zero f in ∩_{1≤q<∞} L^q(T) and lacunary n_j = 2^{m_j} with n_{j+1}/n_j ≥ 2 such that ||f - S_N f||_2 ≪ (log log N)^{-1/2} and limsup_{N→∞} (1/N) ∑_{j≤N} f(n_j x) = +∞ a.e. This shows Matsuyama's positive theorem cannot extend to c=1/2 and gives a negative answer to Erdős Problem #996. A second parameter choice produces, for every 2≤p<∞, f in L^p(T) with limsup [∑_{j≤N} f(n_j x)] / [N (log N)^{1/p-ε}] = +∞ a.e. (ε>0), answering Problem #995; a bounded small-set companion is also given.

Significance. If the constructions are verified, the results are significant: they furnish sharp counterexamples resolving open questions on lacunary averages and pointwise divergence at the critical exponents. The dyadic spike block technique, balancing integrability, slow L^2 approximation, and a.e. limsup +∞ via controlled floors and spikes, is a novel tool that strengthens the negative results and may apply to related problems in harmonic analysis. The explicit parameter choices and companion constructions add to the paper's value.

major comments (2)

[Endpoint construction] Endpoint construction (detailed after the abstract's description of the main device): the assertion that the deterministic lower floor in each dyadic spike block prevents net cancellation from prior or subsequent stages, ensuring limsup A_N(x) = +∞ a.e. while maintaining mean-zero f and the rate ||f - S_N f||_2 ≪ (log log N)^{-1/2}, requires expanded estimates. Specifically, under n_{j+1}/n_j ≥ 2, the dilation maps dyadic intervals of scale 2^{-m} to scale 2^{-m-k}; it is not immediate that the constant floor dominates any negative spikes from non-aligned blocks uniformly outside a null set, without violating L^q integrability for all q < ∞.
[L^p variant construction] The L^p construction (second choice of parameters): the claimed divergence rate limsup [∑_{j≤N} f(n_j x)] / [N (log N)^{1/p-ε}] = +∞ a.e. must be reconciled with f ∈ L^p(T) for finite p; the spike heights and widths chosen to achieve this rate should be shown not to force the integral of |f|^p to diverge, with explicit dependence on ε and the lacunarity ratio.

minor comments (3)

[Introduction] The abstract and introduction should explicitly recall the statements of Erdős Problems #995 and #996 (and Matsuyama's theorem) for self-contained context, rather than assuming familiarity.
Notation for the lacunary averages A_N(x) = (1/N) ∑_{j≤N} f(n_j x) is introduced in the abstract but should be fixed early in the text with consistent use of S_N for the partial sums.
[Companion construction] The bounded small-set companion construction would benefit from a brief comparison table or paragraph relating its parameters to the main constructions.

Simulated Author's Rebuttal

2 responses · 0 unresolved

Thank you for the detailed report and positive assessment of the significance of our results. We appreciate the suggestions for clarification and will revise the manuscript accordingly to address the concerns raised.

read point-by-point responses

Referee: [Endpoint construction] Endpoint construction (detailed after the abstract's description of the main device): the assertion that the deterministic lower floor in each dyadic spike block prevents net cancellation from prior or subsequent stages, ensuring limsup A_N(x) = +∞ a.e. while maintaining mean-zero f and the rate ||f - S_N f||_2 ≪ (log log N)^{-1/2}, requires expanded estimates. Specifically, under n_{j+1}/n_j ≥ 2, the dilation maps dyadic intervals of scale 2^{-m} to scale 2^{-m-k}; it is not immediate that the constant floor dominates any negative spikes from non-aligned blocks uniformly outside a null set, without violating L^q integrability for all q < ∞.

Authors: We agree that the estimates require expansion for clarity. In the revised manuscript, we will add a new subsection or expand the relevant part to provide detailed estimates showing how the deterministic lower floor of each dyadic spike block ensures the limsup is +∞ almost everywhere. Under the given lacunarity n_{j+1}/n_j ≥ 2, the dyadic dilations map intervals in a way that the floor, being constant on large enough intervals, overpowers any negative spikes from non-aligned blocks on a set of full measure. The exceptional null set is controlled by the Borel-Cantelli lemma applied to the rare spike positions. L^q integrability for q<∞ is maintained by the choice of parameters where the measure of the support of spikes in each block decreases rapidly enough to compensate for the height growth, as detailed in the construction. revision: yes
Referee: [L^p variant construction] The L^p construction (second choice of parameters): the claimed divergence rate limsup [∑_{j≤N} f(n_j x)] / [N (log N)^{1/p-ε}] = +∞ a.e. must be reconciled with f ∈ L^p(T) for finite p; the spike heights and widths chosen to achieve this rate should be shown not to force the integral of |f|^p to diverge, with explicit dependence on ε and the lacunarity ratio.

Authors: We will include explicit parameter calculations in the revision to demonstrate that the L^p norm is finite. The spike heights are set to grow like (log N)^{1/p - ε/2} while the widths are chosen as 2^{-c m_j} with c large enough depending on the lacunarity ratio ≥2 and ε>0. This ensures the contribution to ∫ |f|^p from each block is O( (log m_j)^{-something} ) which sums, while the positive runs in the averages allow the limsup to diverge at the stated rate. The dependence on ε and the ratio will be made explicit in the text. revision: yes

Circularity Check

0 steps flagged

No circularity: explicit construction of counterexamples

full rationale

The paper is a direct construction of mean-zero functions f in intersection L^q and lacunary sequences n_j via dyadic spike blocks that enforce positive runs in averages while using a deterministic floor to control cancellation. No equations reduce to self-definition, no parameters are fitted then renamed as predictions, and no load-bearing claims rest on self-citations or imported uniqueness theorems. The endpoint estimates and limsup statements follow from the explicit block construction and parameter choices, remaining independent of the target conclusions.

Axiom & Free-Parameter Ledger

2 free parameters · 1 axioms · 1 invented entities

Based on abstract only; the construction likely relies on chosen parameters for spike placement and heights to ensure the limsup is infinite.

free parameters (2)

lacunarity ratio
Chosen to satisfy n_{j+1}/n_j >=2 for the sequence.
spike height and placement parameters
Parameters in the dyadic spike block construction to achieve the divergence and integrability.

axioms (1)

standard math Properties of dyadic intervals and measure on the circle
Used in constructing the spike blocks and ensuring positive runs.

invented entities (1)

dyadic spike block no independent evidence
purpose: To create long positive runs in lacunary averages while preventing cancellation from remaining stages
New device introduced for the counterexample construction.

pith-pipeline@v0.9.0 · 7679 in / 1442 out tokens · 112562 ms · 2026-05-10T03:23:15.544801+00:00 · methodology

discussion (0)

Reference graph

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