A Construction for Difference Sets with Local Properties

2018/12/18 by Fish, Sara, Lund, Ben, Sheffer, Adam
#Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1812.07651

Abstract

We construct finite sets of real numbers that have a small difference set and strong local properties. In particular, we construct a set A of n real numbers such that |A-A|=n^{log₂ 3} and that every subset A'⊆ A of size k satisfies |A'-A'|≥ k^{log₂ 3}. This construction leads to the first non-trivial upper bound for the problem of distinct distances with local properties.

Related

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• Bounds for the local properties problem for difference sets
• Local Properties in Colored Graphs, Distinct Distances, and Difference\n Sets
• Exponents in the local properties problem for difference sets have a gap at 2
• Difference sets in ℝ^d