PreprintCC BY 4.0arXiv
Simultaneous popular polynomial differences in higher dimensions
Abstract
We extend the simultaneous popular polynomial difference theorem of Conlon, Dong, and the author to polynomial shifts along prescribed directions in 𝔽pD. Let P1,…,Pk∈ℤ[t] be linearly independent polynomials with zero constant terms, and let v1,…,vk∈ℤD∖{0}. We show that, for every ε>0, every sufficiently large prime p, and every A⊆𝔽pD of density α, there exists a nonzero d∈𝔽p such that 𝔼underlinex∈𝔽pD 1A(underlinex) ∏substack1≤ i≤ k ωi=1 1A(underlinex+Pi(d)vi) ≥ α 1+∑i=1kωi−ε simultaneously for every ω∈{0,1}k. The main new ingredient is a directional arithmetic regularity decomposition adapted to mixed directional box norms.
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BibTeX
@misc{hong2026simultaneous,
title = {{Simultaneous popular polynomial differences in higher dimensions}},
author = {Guo-Dong Hong},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.30335},
eprint = {2610.30335},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.30335},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Simultaneous popular polynomial differences in higher dimensions
AU - Guo-Dong Hong
PY - 2026
DA - 2026/10/21
PB - arXiv
DO - 10.48550/arxiv.2610.30335
UR - https://arxiv.org/abs/2610.30335
AB - We extend the simultaneous popular polynomial difference theorem of Conlon, Dong, and the author to polynomial shifts along prescribed directions in 𝔽_p^D. Let P_1,…,P_k∈ℤ[t] be linearly independent polynomials with zero constant terms, and let v_1,…,v_k∈ℤ^D∖{0}. We show that, for every ε>0, every sufficiently large prime p, and every A⊆𝔽_p^D of density α, there exists a nonzero d∈𝔽_p such that 𝔼_underlinex∈𝔽_p^D 1_A(underlinex) ∏_substack1≤ i≤ k ω_i=1 1_A(underlinex+P_i(d)v_i) ≥ α^ 1+∑_i=1^kω_i−ε simultaneously for every ω∈{0,1}^k. The main new ingredient is a directional arithmetic regularity decomposition adapted to mixed directional box norms.
ER -