Better than square-root cancellation and Gaussian behavior on Piatetski-Shapiro Sequences

Abstract

In this paper, we investigate whether Harper's better than square-root cancellation phenomenon still appears in Piatetski-Shapiro sequences, and when one should expect only square-root size. If c=c(X) satisfies (c−1)log X→ 0 as X→∞, then for Steinhaus or Rademacher random multiplicative functions we have 𝔼|∑_{n∈𝒩^(c)(X)}f(n)|=o (√(|𝒩^(c)(X)|)). For Lebesgue almost every fixed 1<c<8/7, we prove the multiplicative energy satisfies E^×(𝒩^(c)(X))=2|𝒩^(c)(X)|²−|𝒩^(c)(X)|+o(|𝒩^(c)(X)|²). Together with a largest prime factor estimate, this allows us to apply the criterion of Soundararajan and Xu and obtain a standard complex Gaussian limit for the normalized Steinhaus sum. To the best of our knowledge, this gives the first asymptotic formula for the multiplicative energy of Piatetski-Shapiro sequences. For fixed exponent argument, our proof combines metric arguments with the Van der Corput's B-process and a double large sieve estimate with Robert-Sargos spacing theorem.

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BibTeX RIS

BibTeX

@misc{zhu2026better,
  title = {{Better than square-root cancellation and Gaussian behavior on Piatetski-Shapiro Sequences}},
  author = {Renjie Zhu and Tianping Zhang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2608.15807},
  eprint = {2608.15807},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.15807},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Better than square-root cancellation and Gaussian behavior on Piatetski-Shapiro Sequences
AU  - Renjie Zhu
AU  - Tianping Zhang
PY  - 2026
DA  - 2026/10/01
PB  - arXiv
DO  - 10.48550/arxiv.2608.15807
UR  - https://arxiv.org/abs/2608.15807
AB  - In this paper, we investigate whether Harper's better than square-root cancellation phenomenon still appears in Piatetski-Shapiro sequences, and when one should expect only square-root size. If c=c(X) satisfies (c−1)log X→ 0 as X→∞, then for Steinhaus or Rademacher random multiplicative functions we have 𝔼|∑_n∈𝒩^(c)(X)f(n)|=o (√(|𝒩^(c)(X)|)). For Lebesgue almost every fixed 1<c<8/7, we prove the multiplicative energy satisfies E^×(𝒩^(c)(X))=2|𝒩^(c)(X)|^2−|𝒩^(c)(X)|+o(|𝒩^(c)(X)|^2). Together with a largest prime factor estimate, this allows us to apply the criterion of Soundararajan and Xu and obtain a standard complex Gaussian limit for the normalized Steinhaus sum. To the best of our knowledge, this gives the first asymptotic formula for the multiplicative energy of Piatetski-Shapiro sequences. For fixed exponent argument, our proof combines metric arguments with the Van der Corput's B-process and a double large sieve estimate with Robert-Sargos spacing theorem.
ER  -

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