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Quantitative bounds in a popular polynomial Szemerédi theorem
- Part of
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- Journal:
- Proceedings of the Royal Society of Edinburgh. Section A: Mathematics , First View
- Published online by Cambridge University Press:
- 05 December 2025, pp. 1-27
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- Article
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We obtain polylogarithmic bounds in the polynomial Szemerédi theorem when the polynomials have distinct degrees and zero constant terms. Specifically, let
$P_1, \dots, P_m \in \mathbb Z[y]$ be polynomials with distinct degrees, each having zero constant term. Then there exists a constant 
$c = c(P_1,\dots,P_m) \gt 0$ such that any subset 
$A \subset \{1,2,\dots,N\}$ of density at least 
$(\log N)^{-c}$ contains a nontrivial polynomial progression of the form 
$x, x+P_1(y), \dots, x+P_m(y)$. In addition, we prove an effective “popular” version, showing that every dense subset 
$A$ has some non-zero 
$y$ such that the number of polynomial progressions in 
$A$ with this difference 
$y$ is asymptotically at least as large as in a random set of the same density as 
$A$.
