Unbounded expansion of polynomials and products

2023/03/28 by Akshat Mudgal, Mudgal, Akshat · 1 citation
Mathematics · #11B13 #11B30 #11B83 #Analytic Number Theory Research #Combinatorics (math.CO) #FOS: Mathematics #Limits and Structures in Graph Theory #Mathematical Dynamics and Fractals #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2303.15910

openalex publication_date 2023/03/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Given d,s ∈ ℕ, a finite set A ⊆ ℤ and polynomials φ₁, …, φ_s ∈ ℤ[x] such that 1 ≤ deg φ_i ≤ d for every 1 ≤ i ≤ s, we prove that |A^(s)| + |φ₁(A) + … + φ_s(A) | ≫_s,d |A|^η_s , for some η_s ≫_d log s / log log s. Moreover if φ_i(0) ≠ 0 for every 1 ≤ i ≤ s, then |A^(s)| + |φ₁(A) … φ_s(A) | ≫_s,d |A|^η_s. These generalise and strengthen previous results of Bourgain--Chang, Pálvölgyi--Zhelezov and Hanson--Roche-Newton--Zhelezov. We derive these estimates by proving the corresponding low-energy decompositions. The latter furnish further applications to various problems of a sum-product flavour, including questions concerning large additive and multiplicative Sidon sets in arbitrary sets of integers.

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