REVIEW
A Counterexample to Wooley's $L^1$ Question for Beatty Sequences
Number Theory · Submitted 1 Nov 2026 · 8 references
Pith's verdict: Chen's construction is correct and settles Wooley's L^1 question with a clean counterexample, showing his (log N)^2 upper bound is sharp.
Why Pith says this
Highlight any text to comment on it.
Referee report
The paper
The paper claims
For one fixed pair of parameters, a Beatty-sequence exponential sum forces the $L^1$ norm to $\gg(\log N)^2$ for infinitely many $N$, disproving Wooley's proposed $\log N$ bound and showing his $(\log 2N)^2$ upper bound is sharp.
The reading
What carries the argument
The argument runs through mechanical words, the zero-one indicators $w_n(\rho)=\lfloor(n+1)\rho+1/2\rfloor-\lfloor n\rho+1/2\rfloor$. With the intercept choice $\beta=-\alpha/2$ these coincide exactly with the Beatty-sequence indicators, so there is no approximation error in passing between the two descriptions. The lower bound uses a rational mechanical word of period $q$: its Fourier polynomial combined with a Dirichlet kernel gives a period-repeated sum, and two logarithmic estimates — an $A_{p,q}(0)\gg\log q$ average (following a Fourier formula of Graham and O'Bryant) and a short-range Dirichlet-kernel bound — stack to give $\gg(\log q)^2$. A continued fraction $\gamma=[0;a_1,a_2,\ldots]$ with $a_{2r+1}=8q_{2r}$ and $a_{2r+2}=2$ then plays the central role: it is simultaneously of finite Diophantine type (indeed $\inf Q^2\|Q\gamma\|>0$), its even convergents have odd denominators $q_{2r}$, and it approximates so well that $|\gamma-p/q|<1/(8q^3)$. Because $q$ is odd, the half-integer-shifted rational arguments stay at distance $\ge1/(2q)$ from integers, and the approximation error stays below $1/(4q)$ for $m\le q^2+1$, so the rational and irrational words agree exactly through the first $q^2$ positions. That exact agreement transfers the rational lower bound to the irrational sequence with no error term.
Extended reading notes
Core claim
The central claim is Theorem 1.1: there are fixed real numbers $\alpha>1$ and $\beta=-\alpha/2$ such that, writing $\gamma=1/\alpha$, we have $\inf_{Q\in\mathbb N}Q^2\|Q\gamma\|>0$, and there is a sequence of positive integers $N_j\to\infty$ with $$\$int_0^{1}$\Bigl|\sum_{\substack{1\le n\le N_j\\n\in\mathcal B(\$\alpha$,\$\beta$)}}e(n\$\theta$)\Bigr|\,d\$\theta$\gg(\log N_j)^2.$$ Since Wooley's theorem gives a uniform $O_{\alpha,\beta}((\log 2N)^2)$ upper bound for such $\gamma$, this construction pins down the true growth rate at the $L^1$ endpoint and settles the proposed logarithmic bound in the negative.
Load-bearing premise
The whole argument pivots on Lemma 4.2: the rational and irrational mechanical words must agree term by term for $n\le q^2$, which demands both that the denominator $q_{2r}$ be odd, so the shifted rational arguments stay at distance $\ge1/(2q)$ from integers, and that $|\gamma-p/q|<1/(8q^3)$, so the error stays below $1/(4q)$; if either condition failed for the chosen $\gamma$, the $(\log N)^2$ rate would not transfer from the rational proxy.
Editorial extensions
If this is right
- Wooley's proposed universal $\ll\log N$ bound is false: some fixed Beatty sequences force the $L^1$ norm to be $\gg(\log N)^2$ for infinitely many $N$.
- The $(\log 2N)^2$ upper bound Wooley proved under finite Diophantine type is optimal: no smaller exponent on $\log N$ is valid in general at the $L^1$ endpoint.
- The failure is isolated at $L^1$: for every fixed $s>1$ the same sequence satisfies $\int_0^1|\sum e(n\theta)|^s\,d\theta\ll N^{s-1}$, so higher moments remain as Wooley found.
- The counterexample lives inside the class where the best-possible upper bound was already known, since $\gamma$ has Diophantine type 2; it shows that class admits no improvement.
- The lower bound holds at the subsequence $N_j=q_{2r}^2$ with one fixed pair $\alpha,\beta$, so the effect is not an artifact of letting the parameters vary with $N$.
Reading between the lines
- The mechanism suggests a natural conjecture: the $L^1$ growth rate tracks the Diophantine type of $\gamma$, with type $k$ plausibly forcing growth $\gg(\log N)^k$ at the endpoint; attempting the same construction with convergents of larger type would test this.
- The exact-agreement trick depends on odd denominators and the half-integer shift $\beta=-\alpha/2$; probing whether another intercept or even-denominator convergents destroy the $(\log N)^2$ bumps would show the effect is specific to this shift.
- The lower bound is proved only at the subsequence $N_j=q_{2r}^2$; whether the $(\log N)^2$ growth persists for all large $N$, rather than only at these scales, is a question the construction leaves open.
Original abstract
For real numbers $\alpha$ and $\beta$, let $\mathcal{B}(\alpha,\beta)=\{\lfloor\alpha m+\beta\rfloor:m\in\mathbb{N}\}$. Wooley asked whether the exponential sum over $\mathcal{B}(\alpha,\beta)\cap[1,N]$ always has $L^1$ norm $O_{\alpha,\beta}(\log N)$. We give a negative answer. We construct fixed $\alpha>1$ and $\beta$, with $1/\alpha$ of finite Diophantine type, such that the $L^1$ norm is $\gg(\log N)^2$ for infinitely many $N$. Wooley also proved that this $L^1$ norm is $O_{\alpha,\beta}((\log(2N))^2)$ when $1/\alpha$ has finite Diophantine type. Our example shows that his upper bound is sharp.
The compiled record
The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
The abstract, sentence by sentence
For real numbers $\alpha$ and $\beta$, let $\mathcal{B}(\alpha,\beta)=\{\lfloor\alpha m+\beta\rfloor:m\in\mathbb{N}\}$. Wooley asked whether the exponential sum over $\mathcal{B}(\alpha,\beta)\cap[1,N]$ always has $L^1$ norm $O_{\alpha,\beta}(\log N)$. We give a negative answer. We construct fixed $\alpha>1$ and $\beta$, with $1/\alpha$ of finite Diophantine type, such that the $L^1$ norm is $\gg(\log N)^2$ for infinitely many $N$. Wooley also proved that this $L^1$ norm is $O_{\alpha,\beta}((\log(2N))^2)$ when $1/\alpha$ has finite Diophantine type. Our example shows that his upper bound is sharp.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | For real numbers $\alpha$ and $\beta$, let $\mathcal{B}(\alpha,\beta)=\{\lfloor\alpha m+\beta\rfloor:m\in\mathbb{N}\}$. | background 1.00 | 0.11 | 0.02 | asserted 0.71 |
| s1 | Wooley asked whether the exponential sum over $\mathcal{B}(\alpha,\beta)\cap[1,N]$ always has $L^1$ norm $O_{\alpha,\beta}(\log N)$. | background 0.96 | 0.03 | 0.03 | asserted 0.71 |
| s2 | We give a negative answer. | result 0.88 | 0.94 | 0.64 | asserted 1.00 |
| s3 | We construct fixed $\alpha>1$ and $\beta$, with $1/\alpha$ of finite Diophantine type, such that the $L^1$ norm is $\gg(\log N)^2$ for infinitely many $N$. | result 0.79 | 0.96 | 0.90 | asserted 1.00 |
| s4 | Wooley also proved that this $L^1$ norm is $O_{\alpha,\beta}((\log(2N))^2)$ when $1/\alpha$ has finite Diophantine type. | background 0.99 | 0.03 | 0.05 | asserted 1.00 |
| s5 | Our example shows that his upper bound is sharp. | interpretation 0.97 | 0.82 | 0.32 | asserted 1.00 |
What was examined, and what was not
- arXiv listing abstract v1
- full text
- figures and tables
- references and citing sentences
- relations to other papers
- P abstract
- S sections
- T body sentences
- C citations
- R cross-paper relations
References
Discussion
Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.