A Maximal Extension of the Best-Known Bounds for the Furstenberg-Sárközy Theorem

2016/12/06 by Rice, Alex · 2 citations
#Classical Analysis and ODEs (math.CA) #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1612.01760

Abstract

We show that if h∈ ℤ[x] is a polynomial of degree k ≥ 2 such that h(ℕ) contains a multiple of q for every q∈ ℕ, known as an intersective polynomial, then any subset of \1,2,…,N\ with no nonzero differences of the form h(n) for n∈ℕ has density at most a constant depending on h and c times (log N)^{-cloglogloglog N}, for any c0, μ=μ(deg(g),deg(h))>0, and μ(2,2)=1/2. We also include a brief discussion of sums of three or more polynomials in the final section.

Cited by

• Furstenberg--Sárközy theorem over number fields
• Notes and computations on forbidden differences

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• Sarkozy's Theorem for P-Intersective Polynomials
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• Furstenberg--Sárközy theorem and partition regularity of polynomial equations over finite fields