Effective bounds for Roth's theorem with shifted square common difference

Ashwin Sah; Mehtaab Sawhney; Sarah Peluse

Effective bounds for Roth's theorem with shifted square common difference

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arxiv 2309.08359 v1 pith:RB42KGTW submitted 2023-09-15 math.NT math.CO

Sarah Peluse , Ashwin Sah , Mehtaab Sawhney This is my paper · ORCID

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keywords absoluteanswersavoidingboundscommonconstantdifferenceeffective

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abstract

Let $S$ be a subset of $\{1,\ldots,N\}$ avoiding the nontrivial progressions $x, x+y^2-1, x+ 2(y^2-1)$. We prove that $|S|\ll N/\log_m{N}$, where $\log_m $ is the $m$-fold iterated logarithm and $m\in\mathbf{N}$ is an absolute constant. This answers a question of Green.

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Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

On polynomial progressions via transference
math.NT 2025-06 conditional novelty 8.0 of 10

For any integer polynomial P with P(0)=0, any subset of [N] avoiding x, x+P(y), ..., x+kP(y) has size at most N (log log log N)^{-c}, with stronger bounds when P'(0)!=0.

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