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A basis of ℤₘ, II

Min Tang, Yong-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.

A basis of Zₘ

Min Tang, Yong-Gao Chen (2006)

Colloquium Mathematicae

Let σ A ( n ) = | ( a , a ' ) A ² : a + a ' = n | , where n ∈ N and A is a subset of N. Erdős and Turán conjectured that for any basis A of order 2 of N, σ A ( n ) is unbounded. In 1990, Imre Z. Ruzsa constructed a basis A of order 2 of N for which σ A ( n ) is bounded in the square mean. In this paper, we show that there exists a positive integer m₀ such that, for any integer m ≥ m₀, we have a set A ⊂ Zₘ such that A + A = Zₘ and σ A ( n ̅ ) 768 for all n̅ ∈ Zₘ.

A note on representation functions with different weights

Zhenhua Qu (2016)

Colloquium Mathematicae

For any positive integer k and any set A of nonnegative integers, let r 1 , k ( A , n ) denote the number of solutions (a₁,a₂) of the equation n = a₁ + ka₂ with a₁,a₂ ∈ A. Let k,l ≥ 2 be two distinct integers. We prove that there exists a set A ⊆ ℕ such that both r 1 , k ( A , n ) = r 1 , k ( A , n ) and r 1 , l ( A , n ) = r 1 , l ( A , n ) hold for all n ≥ n₀ if and only if log k/log l = a/b for some odd positive integers a,b, disproving a conjecture of Yang. We also show that for any set A ⊆ ℕ satisfying r 1 , k ( A , n ) = r 1 , k ( A , n ) for all n ≥ n₀, we have r 1 , k ( A , n ) as n → ∞.

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