A sum of exponentials of the form $f\left(x\right)=exp\left(2\pi i{N}_{1}x\right)+exp\left(2\pi i{N}_{2}x\right)+\cdots +exp\left(2\pi i{N}_{m}x\right)$, where the $N_k$ are distinct integers is called an idempotent trigonometric polynomial (because the convolution of $f$ with itself is $f$) or, simply, an idempotent. We show that for every $p\>1,$ and every set $E$ of the torus $𝕋=ℝ/\mathbb{Z}$ with $\left|E\right|\>0,$ there are idempotents concentrated on $E$ in the $L^p$ sense. More precisely, for each $p\>1,$ there is an explicitly calculated constant $C_p\>0$ so that for each $E$ with $\left|E\right|\>0$ and $ϵ\>0$ one can find an idempotent $f$ such that the ratio ${\left({\int }_E{\left|f\right|}^p/{\int }_{𝕋}{\left|f\right|}^p\right)}^{1/p}$ is greater than $C_p-ϵ$. This is in fact...

For positive integers m and N, we estimate the rational exponential sums with denominator m over the reductions modulo m of elements of the set ℱ(N) = {s/r : r,s ∈ ℤ, gcd(r,s) = 1, N ≥ r > s ≥ 1} of Farey fractions of order N (only fractions s/r with gcd(r,m) = 1 are considered).

Let $\Theta =({\theta }_1,{\theta }_2,{\theta }_3)\in {ℝ}^3$. Suppose that $1,{\theta }_1,{\theta }_2,{\theta }_3$ are linearly independent over $\mathbb{Z}$. For Diophantine exponents $$\begin{array}{cc}\hfill \alpha \left(\Theta \right)& =sup\{\gamma >0:\underset{t\to +\infty }{lim\; sup}{t}^{\gamma }{\psi }_{\Theta }\left(t\right)<+\infty \},\hfill \\ \hfill \beta \left(\Theta \right)& =sup\{\gamma >0:\underset{t\to +\infty }{lim\; inf}{t}^{\gamma }{\psi }_{\Theta }\left(t\right)<+\infty \}\hfill \end{array}$$ we prove $$\beta \left(\Theta \right)\ge \frac{1}{2}\left(\frac{\alpha \left(\Theta \right)}{1-\alpha \left(\Theta \right)}+\sqrt{{\left(\frac{\alpha \left(\Theta \right)}{1-\alpha \left(\Theta \right)}\right)}^2+\frac{4\alpha \left(\Theta \right)}{1-\alpha \left(\Theta \right)}}\right)\alpha \left(\Theta \right).$$

Let $\xi$ be a real number and let $n$ be a positive integer. We define four exponents of Diophantine approximation, which complement the exponents $w_n\left(\xi \right)$ and $w_n^{*}\left(\xi \right)$ defined by Mahler and Koksma. We calculate their six values when $n=2$ and $\xi$ is a real number whose continued fraction expansion coincides with some Sturmian sequence of positive integers, up to the initial terms. In particular, we obtain the exact exponent of approximation to such a continued fraction $\xi$ by quadratic surds.

On montre que les exposants de Lyapunov de l’algorithme de Jacobi-Perron, en dimension $d$ quelconque, sont tous différents et que la somme des exposants extrêmes est strictement positive. En particulier, si $d=2$, le deuxième exposant est strictement négatif.

Dans cet article nous présentons la théorie des équations différentielles $p$-adiques et ses applications concernant le théorème de finitude de la cohomologie $p$-adique d’une variété affine et le théorème de la monodromie $p$-adique des représentations galoisiennes locales.

We prove the rationality of the Łojasiewicz exponent for p-adic semi-algebraic functions without compactness hypothesis. In the parametric case, we show that the parameter space can be divided into a finite number of semi-algebraic sets on each of which the Łojasiewicz exponent is constant.