We study representation functions of asymptotic additive bases and more general subsets of ℕ (sets with few nonrepresentable numbers). We prove that if ℕ∖(A+A) has sufficiently small upper density (as in the case of asymptotic bases) then there are infinitely many numbers with more than five representations in A+A, counting order.

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,...

Let $\varphi (x_1,...,x_h,y)=u_1{x}_1+\cdots +u_h{x}_h+vy$ be a linear form with nonzero integer coefficients $u_1,...,u_h,v.$ Let $𝒜=(A_1,...,A_h)$ be an $h$-tuple of finite sets of integers and let $B$ be an infinite set of integers. Define the representation function associated to the form $\varphi$ and the sets $𝒜$ and $B$ as follows :$$R_{𝒜,B}^{\left(\varphi \right)}\left(n\right)=\text{card}\left(\begin{array}{cc}\hfill \{(a_1,...,a_h,b)\in A_1& \times \cdots \times A_h\times B:\hfill \\ & \varphi (a_1,...,a_h,b)=n\}\hfill \end{array}\right).$$If this representation function is constant, then the set $B$ is periodic and the period of $B$ will be bounded in terms of the diameter of the finite set $\{\varphi (a_1,...,a_h,0):(a_1,...,a_h)\in A_1\times \cdots \times A_h\}.$ Other results for complementing sets with respect to linear forms are also proved.

Soit $c\>1$, $p$ un nombre premier et $𝒜$ une partie de $\mathbb{Z}/p\mathbb{Z}$ de cardinal supérieur à $c\sqrt{p}$ telle que pour tout sous-ensemble non vide $ℬ$ de $𝒜$, on a ${\sum }_{b\in ℬ}b\ne 0$. On montre qu’il existe $s$ premier à $p$ tel que l’ensemble $s.𝒜$ est très concentré autour de l’origine et qu’il est presque entièrement composé d’éléments de partie fractionnaire positive. Plus précisément, on a$$\sum _{a\in 𝒜}∥\frac{sa}{p}∥\<1+O(p^{-1/4}lnp)\text{et}\sum _{\begin{array}{c}a\in 𝒜,\\ \{sa/p\}\ge 1/2\end{array}}∥\frac{sa}{p}∥=O(p^{-1/4}lnp).$$On montre également que les termes d’erreurs ne peuvent être remplacés par $o\left(p^{-1/2}\right)$.