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On a conjecture of Sárközy and Szemerédi

Yong-Gao Chen — 2015

Acta Arithmetica

Two infinite sequences A and B of non-negative integers are called infinite additive complements if their sum contains all sufficiently large integers. In 1994, Sárközy and Szemerédi conjectured that there exist infinite additive complements A and B with lim sup A(x)B(x)/x ≤ 1 and A(x)B(x)-x = O(minA(x),B(x)), where A(x) and B(x) are the counting functions of A and B, respectively. We prove that, for infinite additive complements A and B, if lim sup A(x)B(x)/x ≤ 1, then, for any given M > 1,...

A basis of ℤₘ, II

Min TangYong-Gao Chen — 2007

Colloquium Mathematicae

Given a set A ⊂ ℕ let σ A ( n ) denote the number of ordered pairs (a,a’) ∈ A × A such that a + a’ = n. Erdős and Turán conjectured that for any asymptotic basis A of ℕ, σ A ( n ) is unbounded. We show that the analogue of the Erdős-Turán conjecture does not hold in the abelian group (ℤₘ,+), namely, for any natural number m, there exists a set A ⊆ ℤₘ such that A + A = ℤₘ and σ A ( n ̅ ) 5120 for all n̅ ∈ ℤₘ.

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