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.

Let ${\sigma }_A\left(n\right)=\left|(a,a^{\text{'}})\in A²:a+a^{\text{'}}=n\right|$, 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.

Let $𝒯$ be a system of disjoint subsets of ${\mathbb{N}}^{*}$. 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 ${𝑟}_{𝑛}\left(𝐴\right);{𝑟}_{𝑛}\left(𝐴\right):=\left|\left\{({𝑎}_{𝑖},{𝑎}_{𝑗}):{𝑎}_{𝑖}\<{𝑎}_{𝑗};{𝑎}_{𝑖},{𝑎}_{𝑗}\in 𝐴;𝑛={𝑎}_{𝑖}+{𝑎}_{𝑗}\right\}\right|$, for all sufficiently large $n\in T_j$ and $j\in {\mathbb{N}}^{*}$ A theorem of P. Erdös is generalized.

Let A be a multiplicative subgroup of $\mathbb{Z}{*}_p$. 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 $6A\supseteq \mathbb{Z}{*}_p$ for $\left|A\right|>p^{11/23+ϵ}$. In addition, we extend a result of Shkredov to show that ${\left|2A\right|\gg \left|A\right|}^{8/5-ϵ}$ for $\left|A\right|\ll p^{5/9}$.

Nous améliorons les meilleures bornes supérieures et inférieures connues pour la fonction $X$ d’Erdös et Graham définie par $X\left(h\right)={max}_{h𝒜\sim \mathbb{N}}{max}_{a\in {𝒜}^{*}}{\mathrm{ord}}^{*}\left(𝒜\setminus a\right)$, 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...