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Documents  11B83 | enregistrements trouvés : 3

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Given a subset A of an additive group, how small can the sumset $A+A = \lbrace a+a' : a, a' \epsilon$ $A \rbrace$ be ? And what can be said about the structure of $A$ when $A + A$ is very close to the smallest possible size ? The aim of this talk is to partially answer these two questions when A is either a subset of $\mathbb{Z}$, $\mathbb{Z}/n\mathbb{Z}$, $\mathbb{R}$ or $\mathbb{T}$ and to explain how in this problem discrete and continuous setting are linked. This should also illustrate two important principles in additive combinatorics : reduction and rectification.
This talk is partially based on some joint work with Pablo Candela and some other work with Paul Péringuey.
Given a subset A of an additive group, how small can the sumset $A+A = \lbrace a+a' : a, a' \epsilon$ $A \rbrace$ be ? And what can be said about the structure of $A$ when $A + A$ is very close to the smallest possible size ? The aim of this talk is to partially answer these two questions when A is either a subset of $\mathbb{Z}$, $\mathbb{Z}/n\mathbb{Z}$, $\mathbb{R}$ or $\mathbb{T}$ and to explain how in this problem discrete and continuous ...

11B13 ; 11B83 ; 11B75

Let $s(m)$ denote the number of distinct powers of 2 in the binary representation of $m$. Thus the Thue-Morse sequence is $(-1)^{s(m)}$ and
$T_n(x)=\sum_{0\leq m< 2^n}(-1)^{s(m)}e(mx)=\prod_{0\leq r< n}(1-e(2^rx))$
is a trigonometric generating generating function of the sequence. The work of Mauduit and Rivat on $(-1)^{s(p)}$ depends on nontrivial bounds for $\left \| T_n \right \|_1$ and for $\left \| T_n \right \|_\infty $. We consider other norms of the $T_n$. For positive integers $k$ let
$M_k(n)=\int_{0}^{1}\left | T_n(x) \right |^{2k}dx$
We show that the sequence $M_k(n)$ satisfies a linear recurrence of order $k$. Moreover, we determine a $k\times k$ matrix whose characteristic polynomial determines this linear recurrence.
This is joint work with Mauduit and Rivat.
Let $s(m)$ denote the number of distinct powers of 2 in the binary representation of $m$. Thus the Thue-Morse sequence is $(-1)^{s(m)}$ and
$T_n(x)=\sum_{0\leq m< 2^n}(-1)^{s(m)}e(mx)=\prod_{0\leq r< n}(1-e(2^rx))$
is a trigonometric generating generating function of the sequence. The work of Mauduit and Rivat on $(-1)^{s(p)}$ depends on nontrivial bounds for $\left \| T_n \right \|_1$ and for $\left \| T_n \right \|_\infty $. We consider oth...

11B83

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