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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.

Integers

### A class of symmetric 2-bases.

Mathematica Scandinavica

### A curious bijection on natural numbers.

Journal of Integer Sequences [electronic only]

### A generalization of a theorem of Erdös on asymptotic basis of order $2$

Journal de théorie des nombres de Bordeaux

Let $𝒯$ be a system of disjoint subsets of ${ℕ}^{*}$. In this paper we examine the existence of an increasing sequence of natural numbers, $A$, that is an asymptotic basis of all infinite elements ${T}_{j}$ of $𝒯$ simultaneously, satisfying certain conditions on the rate of growth of the number of representations ${𝑟}_{𝑛}\left(𝐴\right);{𝑟}_{𝑛}\left(𝐴\right):=\left|\left\{\left({𝑎}_{𝑖},{𝑎}_{𝑗}\right):{𝑎}_{𝑖}<{𝑎}_{𝑗};{𝑎}_{𝑖},{𝑎}_{𝑗}\in 𝐴;𝑛={𝑎}_{𝑖}+{𝑎}_{𝑗}\right\}\right|$, for all sufficiently large $n\in {T}_{j}$ and $j\in {ℕ}^{*}$ A theorem of P. Erdös is generalized.

Acta Arithmetica

Integers

Acta Arithmetica

### A note on sumsets of subgroups in $ℤ{*}_{p}$

Acta Arithmetica

Let A be a multiplicative subgroup of $ℤ{*}_{p}$. Define the k-fold sumset of A to be $kA={x}_{1}+...+{x}_{k}:{x}_{i}\in A,1\le i\le k$. We show that $6A\supseteq ℤ{*}_{p}$ for $|A|>{p}^{11/23+ϵ}$. In addition, we extend a result of Shkredov to show that ${|2A|\gg |A|}^{8/5-ϵ}$ for $|A|\ll {p}^{5/9}$.

### A note on the postage stamp problem.

Journal of Integer Sequences [electronic only]

Acta Arithmetica

### A propos de la fonction $X$ d’Erdös et Graham

Annales de l’institut Fourier

Nous améliorons les meilleures bornes supérieures et inférieures connues pour la fonction $X$ d’Erdös et Graham définie par $X\left(h\right)={max}_{h𝒜\sim ℕ}\phantom{\rule{4pt}{0ex}}{max}_{a\in {𝒜}^{*}}\phantom{\rule{4pt}{0ex}}{\mathrm{ord}}^{*}\left(𝒜\setminus a\right)$, où le premier maximum est pris sur toutes les bases (exactes) $𝒜$ d’ordre au plus $h$, où ${𝒜}^{*}$ désigne le sous-ensemble de $𝒜$ composé des éléments $a$ tels que $𝒜\setminus \left\{a\right\}$ soit encore une base et où, enfin, ${\mathrm{ord}}^{*}\left(𝒜\right)$ désigne l’ordre (exact) de $𝒜$. Notre étude nous conduira, entre autres, à prouver un nouveau résultat additif général découlant de la méthode isopérimétrique et à étudier trois problèmes additifs...

Acta Arithmetica

Acta Arithmetica

Acta Arithmetica

Acta Arithmetica

### Abschätzung der Dichte von Summenmengen.

Mathematische Zeitschrift

### Addition theorems in F q [x].

Seminaire de Théorie des Nombres de Bordeaux

Acta Arithmetica

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