Arithmetic Structure in Sparse Difference Sets

2010/04/15 by Mariah Hamel, Neil Lyall, Hamel, Mariah +5
Mathematics · #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT) #math.CO #math.NT

paper · pdf · doi:10.48550/arxiv.1004.2723

7 pages, to appear in the Journal of Number Theory.

arxiv created 2010/04/15 · arxiv updated 2010/04/19

Abstract

Using a slight modification of an argument of Croot, Ruzsa and Schoen we establish a quantitative result on the existence of a dilated copy of any given configuration of integer points in sparse difference sets. More precisely, given any configuration \v₁,...,v_ℓ\ of vectors in ℤ^d, we show that if A⊂[1,N]^d with |A|/N^d≥ C N^-1/ℓ, then there necessarily exists r≠0 such that \rv₁, ...,rv_ℓ\⊆ A-A.

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