On the Quantitative Bounds of Nonlinear Roth Theorem
Abstract
We prove that any subset A ⊆ {1, …, N} lacking non-trivial polynomial progressions (x, x + y, x + y2) satisfies lvert A rvert ≤ CN exp(−c(log log N)β) for effective absolute constants C, c, β> 0. The argument develops a density increment framework localized to anisotropic quadratic Bohr sets. To handle mixed linear and quadratic frequencies, we establish a multilinear transference principle coupled with a relative quadratic inverse theorem based on Balog--Szemer{é}di--Gowers additive energy rigidity. Boundary stability across increments is maintained via semi-algebraic model regularization using cylindrical algebraic decomposition and Cauchy--Crofton estimates, while dimension growth is controlled by dissociated basis reduction and multidimensional Erd{ő}s--Tur{á}n--Koksma equidistribution.
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BibTeX
@misc{zhao2026quantitative,
title = {{On the Quantitative Bounds of Nonlinear Roth Theorem}},
author = {Yunfan Zhao},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.13204},
eprint = {2610.13204},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.13204},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On the Quantitative Bounds of Nonlinear Roth Theorem
AU - Yunfan Zhao
PY - 2026
DA - 2026/10/12
PB - arXiv
DO - 10.48550/arxiv.2610.13204
UR - https://arxiv.org/abs/2610.13204
AB - We prove that any subset A ⊆ {1, …, N} lacking non-trivial polynomial progressions (x, x + y, x + y^2) satisfies lvert A rvert ≤ CN exp(−c(log log N)^β) for effective absolute constants C, c, β> 0. The argument develops a density increment framework localized to anisotropic quadratic Bohr sets. To handle mixed linear and quadratic frequencies, we establish a multilinear transference principle coupled with a relative quadratic inverse theorem based on Balog--Szemer{é}di--Gowers additive energy rigidity. Boundary stability across increments is maintained via semi-algebraic model regularization using cylindrical algebraic decomposition and Cauchy--Crofton estimates, while dimension growth is controlled by dissociated basis reduction and multidimensional Erd{ő}s--Tur{á}n--Koksma equidistribution.
ER -