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### A basis of ℤₘ, II

Colloquium Mathematicae

Given a set A ⊂ ℕ let ${\sigma }_{A}\left(n\right)$ 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, ${\sigma }_{A}\left(n\right)$ 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 ${\sigma }_{A}\left(n̅\right)\le 5120$ for all n̅ ∈ ℤₘ.

### A basis of Zₘ

Colloquium Mathematicae

Let ${\sigma }_{A}\left(n\right)=|\left(a,{a}^{\text{'}}\right)\in A²:a+{a}^{\text{'}}=n|$, where n ∈ N and A is a subset of N. Erdős and Turán conjectured that for any basis A of order 2 of N, ${\sigma }_{A}\left(n\right)$ is unbounded. In 1990, Imre Z. Ruzsa constructed a basis A of order 2 of N for which ${\sigma }_{A}\left(n\right)$ 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 ${\sigma }_{A}\left(n̅\right)\le 768$ for all n̅ ∈ Zₘ.

Acta Arithmetica

Integers

Integers

### A note on representation functions with different weights

Colloquium Mathematicae

For any positive integer k and any set A of nonnegative integers, let ${r}_{1,k}\left(A,n\right)$ 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 ${r}_{1,k}\left(A,n\right)={r}_{1,k}\left(ℕ\setminus A,n\right)$ and ${r}_{1,l}\left(A,n\right)={r}_{1,l}\left(ℕ\setminus A,n\right)$ 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 ${r}_{1,k}\left(A,n\right)={r}_{1,k}\left(ℕ\setminus A,n\right)$ for all n ≥ n₀, we have ${r}_{1,k}\left(A,n\right)\to \infty$ as n → ∞.

Acta Arithmetica

### A Rademacher type formula for partitions and overpartitions.

International Journal of Mathematics and Mathematical Sciences

Acta Arithmetica

Acta Arithmetica

Acta Arithmetica

Acta Arithmetica

Integers

### Characterizing Frobenius semigroups by filtration.

Journal of Integer Sequences [electronic only]

Acta Arithmetica

### Even partition functions.

Séminaire Lotharingien de Combinatoire [electronic only]

### Every function is the representation function of an additive basis for the integers.

Portugaliae Mathematica. Nova Série

Acta Arithmetica

Integers

Integers

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