Nous donnons une représentation géométrique des suites doubles uniformément récurrentes de fonction de complexité rectangulaire $mn+n$. Nous montrons que ces suites codent l’action d’une ${\mathbb{Z}}^2$-action définie par deux rotations irrationnelles sur le cercle unité. La preuve repose sur une étude des suites doubles dont les lignes sont des suite sturmiennes de même langage.

Les ensembles “propres” pour une suite de Sidon sont caractérisés par une propriété de convergence des séries lacunaires à spectre dans la suite.

Let $n>m\ge 2$ be positive integers and $n=(m+1)\ell +r$, where $0\le r\le m.$ Let $C$ be a subset of $\{0,1,\cdots ,n\}$. We prove that if $$\left|C\right|>\left\{\begin{array}{cc}\lfloor n/2\rfloor +1\hfill & \text{if}m\text{is}\text{odd},\hfill \\ m\ell /2+\delta \hfill & \text{if}m\text{is}\text{even},\hfill \end{array}\right.$$ where $\lfloor x\rfloor$ denotes the largest integer less than or equal to $x$ and $\delta$ denotes the cardinality of even numbers in the interval $[0,min\{r,m-2\left\}\right]$, then $C-C$ contains a power of $m$. We also show that these lower bounds are best possible.

We call a subset A of an abelian group G sum-dominant when |A+A| > |A-A|. If |A⨣A| > |A-A|, where A⨣A comprises the sums of distinct elements of A, we say A is restricted-sum-dominant. In this paper we classify the finite abelian groups according to whether or not they contain sum-dominant sets (respectively restricted-sum-dominant sets). We also consider how much larger the sumset can be than the difference set in this context. Finally, generalising work of Zhao, we provide asymptotic estimates...

In this paper we prove the following theorems in incidence geometry. 1. There is $\delta >0$ such that for any $P_1,\cdots ,P_4$, and $Q_1,\cdots ,Q_n\in {ℂ}^2$, if there are $\le n^{(1+\delta )/2}$ many distinct lines between $P_i$ and $Q_j$ for all $i$, $j$, then $P_1,\cdots ,P_4$ are collinear. If the number of the distinct lines is $<c{n}^{1/2}$ then the cross ratio of the four points is algebraic. 2. Given $c>0$, there is $\delta >0$ such that for any $P_1,P_2,P_3\in {ℂ}^2$ noncollinear, and $Q_1,\cdots ,Q_n\in {ℂ}^2$, if there are $\le c{n}^{1/2}$ many distinct lines between $P_i$ and $Q_j$ for all $i$, $j$, then for any $P\in {ℂ}^2\setminus \{P_1,P_2,P_3\}$, we have $\delta n$ distinct lines between $P$ and $Q_j$. 3. Given $c>0$, there is...