Infinite unrestricted sumsets of the form B+B in sets with large density

2024/04/18 by Kousek, Ioannis, Radić, Tristán
#05D10 #11B13 #11B30 #37A05 #Combinatorics (math.CO) #Dynamical Systems (math.DS) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2404.12201

Abstract

For a set A ⊂ ℕ we characterize in terms of its density when there exists an infinite set B ⊂ ℕ and t ∈ \0,1\ such that B+B ⊂ A-t, where B+B : =\b₁+b₂\colon b₁,b₂ ∈ B\. Specifically, when the lower density \underlined(A) >1/2 or the upper density d(A)> 3/4, the existence of such a set B⊂ ℕ and t∈ \0,1\ is assured. Furthermore, whenever \underlined(A) > 3/4 or d(A)>5/6, we show that the shift t is unnecessary and we also provide examples to show that these bounds are sharp. Finally, we construct a syndetic three-coloring of the natural numbers that does not contain a monochromatic B+B+t for any infinite set B ⊂ ℕ and number t ∈ ℕ.

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