Difference Sets and Polynomials

2015/04/20 by Lyall, Neil, Rice, Alex
#Classical Analysis and ODEs (math.CA) #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1504.04904

Abstract

We provide upper bounds on the largest subsets of \1,2,…,N\ with no differences of the form h₁(n₁)+⋯+h_ℓ(n_ℓ) with n_i∈ ℕ or h₁(p₁)+⋯+h_ℓ(p_ℓ) with p_i prime, where h_i∈ ℤ[x] lie in in the classes of so-called intersective and P-intersective polynomials, respectively. For example, we show that a subset of \1,2,…,N\ free of nonzero differences of the form n^j+m^k for fixed j,k∈ ℕ has density at most e^{-(log N)μ} for some μ=μ(j,k)>0. Our results, obtained by adapting two Fourier analytic, circle method-driven strategies, either recover or improve upon all previous results for a single polynomial. UPDATE: While the results and proofs in this preprint are correct, the main result (Theorem 1.1) has been superseded prior to publication by a new paper ( https://arxiv.org/abs/1612.01760 ) that provides better results with considerably less technicality, to which the interested reader should refer.

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