Évènements

Let $X=G/K$ be a Riemannian symmetric space of non compact type with real rank one. For $\lambda \in \mathbb{C}$ and $f$ an integrable function on the Furstenberg boundary $K/M$, the Poisson transform $P_\lambda$ of $f$ is given by

$
(P_\lambda f)(x)=\int_{K/M} e^{(i\lambda+\rho)A(x,b)}f(b)db, \quad \mbox{for} \; x\in X.
$

The aim of this talk is to present a necessary and a suffucient condition on eigenfunctions of the Laplace-Beltrami operator associated to $X$ with eigenvalue $-(\lambda^2+\rho^2)$ to have an $L^p$-Poisson integral representations on the boundary $K/M$. A special discuss of the case of the exceptional symmetric space.

We prove the following structural result, resembling the Arithmetical Regularity Lemma of B. Green, and Graph Container Theorem in hypergraphs:
Lemma: Let $A_1,A_2,\ldots,A_k\subset\mathbb{F}_p$ be such that $|A_i| \gg p$ for all $i$. Assume that $(A_1 * A_2 * \ldots * A_k)(a) = o(p^{k-1})$ for some $a \in \mathbb{F}_p$.
Then there exist sets $W_1, \ldots, W_k$, which we call wrappers, and sets $Y_1, \ldots, Y_k$, such that:
$(W_1 * W_2 * \ldots * W_k)(b) = o(p^{k-1})$ for some $b \in \mathbb{F}_p$ , $A_i \setminus Y_i \subseteq W_i$ and $|Y_i| = o(p)$ for all $i$, $|W_i|_{\omega} = p^{o(1)}$ for all $i$, where $|\cdot|_{\omega}$ is a Wiener norm.
As a consequence of wrappers having a small Wiener norm, we obtain the following results.
If $A(A+A)$ does not cover all nonzero residues in $\mathbb{F}_p$, then $|A| \leqslant p/8 + o(p)$.
If $A$ is both sum-free and satisfies $A = A^*$, then $|A| \leqslant p/9 + o(p)$.
If $|A| \gg \frac{\log\log{p}}{\sqrt{\log{p}}}p$, then $|A + A^*| \geqslant (1 – o(1))\min(2\sqrt{|A|p},p)$.
Constants 1/8, 1/9, and 2 are optimal.
To obtain this result, we use Croot-Laba-Sisask Lemma and properties of Wiener norms.
This continues the work of A. Balog, K. Benjamin, P.-Y. Bienvenu, K. Broughan, F. Hennecart, B. Murphy, M. Rudnev, I. Shkredov, I. Shparlinski, and E. Yazici.