PreprintCC BY-NC-ND 4.0arXiv
The Furstenberg-Sárközy theorem for sums of an even number of odd powers
Abstract
We obtain a Furstenberg-Sárközy-type result for sets A⊂ [N] whose difference set A−A does not contain the sum of s-many k-th powers of positive integers, with k>1 odd and s>0 even. Namely, we prove that such sets must satisfy a power-saving bound |A| ≪ N1−1/kmin{s σk, 1/2}+ε for any fixed ε>0, where $σ_k >0 $ is any admissible saving in a classical one-variable Weyl estimate. In particular, we can take σk=max{21−k, 1/(k(k−1))} using the classical theory and the best currently available bounds for classical Weyl sums. A greedy construction produces a set A⊂[N] with |A|≫ N1−s/k for which A−A contains no sum of s-many positive k-th powers, so our power-saving bound is of the correct shape.
Related
- Extensions of the Furstenberg-Sárközy theorem via the arithmetic level-d inequality
- An explicit power-saving bound for Sárközy's Theorem on shifted primes, assuming GRH
- A Generalization of Sárközy's theorem in function fields
- Matrices over non-commutative rings as sums of fifth and seventh powers
- Local obstructions and certified computations for perfect powers in power sums
BibTeX
@misc{kalogirou2026furstenbergsárközy,
title = {{The Furstenberg-Sárközy theorem for sums of an even number of odd powers}},
author = {Alexandros Kalogirou and Andrew Lott and Ákos Magyar and Akash Singha Roy},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.16595},
eprint = {2609.16595},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.16595},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - The Furstenberg-Sárközy theorem for sums of an even number of odd powers
AU - Alexandros Kalogirou
AU - Andrew Lott
AU - Ákos Magyar
AU - Akash Singha Roy
PY - 2026
DA - 2026/09/15
PB - arXiv
DO - 10.48550/arxiv.2609.16595
UR - https://arxiv.org/abs/2609.16595
AB - We obtain a Furstenberg-Sárközy-type result for sets A⊂ [N] whose difference set A−A does not contain the sum of s-many k-th powers of positive integers, with k>1 odd and s>0 even. Namely, we prove that such sets must satisfy a power-saving bound |A| ≪ N^1−1/kmin{s σ_k, 1/2}+ε for any fixed ε>0, where $σ_k >0 $ is any admissible saving in a classical one-variable Weyl estimate. In particular, we can take σ_k=max{2^1−k, 1/(k(k−1))} using the classical theory and the best currently available bounds for classical Weyl sums. A greedy construction produces a set A⊂[N] with |A|≫ N^1−s/k for which A−A contains no sum of s-many positive k-th powers, so our power-saving bound is of the correct shape.
ER -