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A Counterexample to Wooley's L1 Question for Beatty Sequences
Abstract
For real numbers α and β, let ℬ(α,β)={⌊αm+β⌋:m∈ℕ}. Wooley asked whether the exponential sum over ℬ(α,β)∩[1,N] always has L1 norm Oα,β(log N). We give a negative answer. We construct fixed α>1 and β, with 1/α of finite Diophantine type, such that the L1 norm is ≫(log N)2 for infinitely many N. Wooley also proved that this L1 norm is Oα,β((log(2N))2) when 1/α has finite Diophantine type. Our example shows that his upper bound is sharp.
Related
BibTeX
@misc{chen2026counterexample,
title = {{A Counterexample to Wooley's $L^1$ Question for Beatty Sequences}},
author = {Dijia Chen},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.01494},
eprint = {2611.01494},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.01494},
}
RIS
TY - UNPB
TI - A Counterexample to Wooley's L^1 Question for Beatty Sequences
AU - Dijia Chen
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.01494
UR - https://arxiv.org/abs/2611.01494
AB - For real numbers α and β, let ℬ(α,β)={⌊αm+β⌋:m∈ℕ}. Wooley asked whether the exponential sum over ℬ(α,β)∩[1,N] always has L^1 norm O_α,β(log N). We give a negative answer. We construct fixed α>1 and β, with 1/α of finite Diophantine type, such that the L^1 norm is ≫(log N)^2 for infinitely many N. Wooley also proved that this L^1 norm is O_α,β((log(2N))^2) when 1/α has finite Diophantine type. Our example shows that his upper bound is sharp.
ER -