On polynomial progressions via transference

2025/06/16 by Altman, Daniel, Sawhney, Mehtaab
#Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2506.13010

Abstract

We prove new cases of reasonable bounds for the polynomial Szemerédi theorem both over ℤ/Nℤ with N prime and over the integers. In particular, we prove reasonable bounds for Szemerédi's theorem in the integers with fixed polynomial common difference. That is, we prove for any polynomial P(y)∈ ℤ[y] with P(0) = 0, that the largest subset A⊆ [N] avoiding the pattern x, x+P(y),…, x+ kP(y) has size bounded by ≪_P,kN(logloglog N)^-Ω_P,k(1).

Citations

• Primes of the form p² + nq²
• Quantitative concatenation for polynomial box norms
• Corners with polynomial side length
• Quasipolynomial bounds on the inverse theorem for the Gowers U^s+1[N]-norm
• Improved bounds for five-term arithmetic progressions
• Strong Bounds for 3-Progressions
• A non-flag arithmetic regularity lemma and counting lemma
• On several notions of complexity of polynomial progressions
• Further bounds in the polynomial Szemerédi theorem over finite fields
• Quantitative bounds in the nonlinear Roth theorem
• Quantitative bounds in the inverse theorem for the Gowers\n U^s+1-norms over cyclic groups
• Improved estimates for polynomial Roth type theorems in finite fields
• Periodic nilsequences and inverse theorems on cyclic groups
• The quantitative behaviour of polynomial orbits on nilmanifolds
• New bounds for Szemeredi's theorem, II: A new bound for r₄(N)
• Linear Equations in Primes
• Roth's theorem in the primes

Related

• Further bounds in the polynomial Szemerédi theorem over finite fields
• A Quantitative Bound For Szemerédi's Theorem for a Complexity One Polynomial Progression over ℤ/Nℤ
• Quantitative bounds in a popular polynomial Szemerédi theorem
• Lower bounds in the polynomial Szemerédi theorem
• Intersective polynomials and polynomial Szemeredi theorem