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Michel Weber

Publications and source records attributed to Michel Weber.

At least 37 records · Page 2Linked to original sources
arXiv2014

Convergence of series of dilated functions and spectral norms of GCD matrices

We establish a connection between the $L^2$ norm of sums of dilated functions whose $j$th Fourier coefficients are $\mathcal{O}(j^{-α})$ for some $α\in (1/2,1)$, and the spectral norms of certain greatest common divisor (GCD) matrices. Utilizing recent bounds for these spectral norms, we obtain sharp conditions for the convergence in $L^2$ and for the almost everywhere convergence of series of dilated functions.

math.CA↗
arXiv2014

$\boldsymbol L^{\boldsymbol 1}$-Norm of Steinhaus chaose on the polydisc

Let $J_n\subset[1,n]$, $n=1,2,\ldots$ be increasing sets of mutually coprime numbers. Under reasonable conditions on the coefficient sequence $\{c^j_n\}_{n,j}$, we show that $$ \lim_{T\to \infty}\frac{1}{T} \int_{0}^T \Big| \sum_{j\in J_n} c^j_n\,j^{it}\Big| \dd t\sim \big(\fracπ {2}\sum_{j\in J_n} (c^j_n)^2\big)^{1/2} $$ as $n\to \infty$. We also show by means of an elementary device that for all $0<\a<2$, \begin{eqnarray*} \lim_{T\to \infty} \Big(\frac{1}{T} \int_{0}^T \big| \sum_{n=1}^N n^{-it}\big|^\a\dd t\Big)^{1/\a} \ge C_\a\, \frac{ N^{\frac{1}{2}}} {\big( \log N\big)^{{\frac{1}{\a} -\frac{1}{2} }}}. \end{eqnarray*} The proof uses Ayyad, Cochrane and Zheng estimate on the number of solutions of the equation $x_1x_2=x_3x_4$. In the case $\a=1$, this approaches Helson's bound up to a factor $(\log N)^{1/4}$.

math.CV↗
arXiv2013

On series $\sum c_k f(kx)$ and Khinchin's conjecture

We prove the optimality of a criterion of Koksma (1953) in Khinchin's conjecture, settling a long standing open problem in analysis. Using this result, we also give a near optimal condition for the a.e.\ convergence of series $\sum_{k=1}^\infty c_k f(kx)$ for $f\in L^2$.

math.CA↗
arXiv2012

On series of dilated functions

Given a periodic function $f$, we study the almost everywhere and norm convergence of series $\sum_{k=1}^\infty c_k f(kx)$. As the classical theory shows, the behavior of such series is determined by a combination of analytic and number theoretic factors, but precise results exist only in a few special cases. In this paper we use connections with orthogonal function theory and GCD sums to prove several new results, improve old ones and also to simplify and unify the theory.

math.CA↗
arXiv2012

Local Suprema of Dirichlet Polynomials and Zerofree Regions of the Riemann Zeta-Function

A new zerofree region of the Riemann Zeta-function $ζ$ is identified by using Turán's localization criterion linking zeros of $ζ$ with uniform local suprema of sets of Dirichlet polynomials expanded over the primes. The proof is based on a randomization argument. An estimate for local extrema for some finite families of shifted Dirichlet polynomials, % associated to linearly independent sequences of reals is established by preliminary considering their local increment properties, by means of Montgomery-Vaughan's variant of Hilbert's inequality. A covering argument combined with Turán's localization criterion allows to conclude.

math.NT↗
arXiv2011

A bound for Mean values of Fourier transforms

We show that there exists a sequence $\{n_k, k\ge 1\}$ growing at least geometrically such that for any finite non-negative measure $ν$ such that $\hat ν\ge 0$, any $T>0$, $$ \int_{-2^{n_k} T}^{2^{n_k} T} \hat ν(x) \dd x \ll_\e T\,2^{2^{(1+\e)n_k}} \int_\R \Big|{\sin {xT} \over xT} \Big|^{n_k^2} ν(\dd x). $$

math.FA↗
arXiv2010

On Small deviations of Gaussian processes using majorizing measures

We give two examples of periodic Gaussian processes, having entropy numbers of exactly same order but radically different small deviations. Our construction is based on classical Knopp's result yielding of existence of continuous nowhere differentiable functions, and more precisely on Loud's functions. We also obtain a general lower bound for small deviations using the majorizing measure method. We show on examples that our bound is sharp. We also apply it to Gaussian independent sequences and to the generic class of ultrametric Gaussian processes. 60F15, 60G50 ; Secondary: 60F05

math.PR↗
arXiv2010

A sharp correlation inequality with an application to almost sure local limit theorem

We prove a new sharp correlation inequality for sums of i.i.d. square integrable lattice distributed random variables. We also apply it to establish an almost sure local limit theorem for iid square integrable random variables taking values in an arbitrary lattice. This extends a recent similar result jointly obtained with Antonini-Giuliano, under a slightly stronger absolute moment assumption (of order $2+u$ with $u>0$). The approach used to treat the case $u>0$ breaks down when $u=0$. MacDonald's concept of the Bernoulli part of a random variable is used in a crucial way to remedy this.

math.PR↗
arXiv2010

Instants of small amplitude of Brownian motion and application to the Kubilius model

Let $W(t), t\ge 0$ be standard Brownian motion. We study the size of the time intervals $I$ which are admissible for the long range of slow increase, namely given a real $z>0$, $$ \sup_{t\in I}{|W(t)|\over \sqrt t} \le z, $$ and we estimate their number of occurences. We obtain optimal results in terms of class test functions and, by means of the quantitative Borel-Cantelli lemma, a fine frequency result concerning their occurences. Using Sakhanenko's invariance principe to transfer the results to the Kubilius model, we derive applications to the prime number divisor function. We obtain refinements of some results recently proved by Ford and Tenenbaum.

math.PR↗
arXiv2009

Dirichlet polynomials: some old and recent results, and their interplay in number theory

In the first part of this expository paper, we present and discuss the interplay of Dirichlet polynomials in some classical problems of number theory, notably the Lindelöf Hypothesis. We review some typical properties of their means and continue with some investigations concerning their supremum properties. Their random counterpart is considered in the last part of the paper, where a analysis of their supremum properties, based on methods of stochastic processes, is developed.

math.NT↗
arXiv2009

A Sharp Estimate for Divisors of Bernoulli Sums

Let $S_n=\e_1+...+\e_n$, where $ \e_i $ are i.i.d. Bernoulli r.v.'s. Let $0\le r_d(n)<2d$ be the least residue of $n$ mod$(2d)$, $\bar r_d(n)= 2d -r_d(n)$ and $\b(n,d)=\max ({1\over d}, {1\over \sqrt n})[e^{- {r_d(n)^2/2 n}} +e^{- {\bar r_d(n)^2/2 n}}]$. We show that $$\sup_{2\le d\le n} \big|¶\big\{d|S_n\big\}- E(n,d) \big|= {\cal O}\big({\log^{5/2} n \over n^{3/2}}\big), $$ where $E(n,d) $ verifies $c_1\b(n,d)\le E(n,d)\le c_2\b(n,d) $ and $c_1,c_2 $ are numerical constants.

math.NT↗
arXiv2009

A general strong law of large numbers for additive arithmetic functions

Let $f(n)$ be a strongly additive complex valued arithmetic function. Under mild conditions on $f$, we prove the following weighted strong law of large numbers: if $ X,X_1,X_2,... $ is any sequence of integrable i.i.d. random variables, then $$ \lim_{N\to \infty} {\sum_{n=1}^N f(n) X_n \over\sum_{n=1}^N f(n)} \buildrel{a.s.}\over{=} \E X . $$

math.NT↗
arXiv2009

A Remark on Zeros of Brownian Motion

Let $ \{W(t), t\ge 0\}$ be a standard Brownian motion. If $I$ is a bounded interval on which $W $ has no zero, an almost sure lower bound to $\inf\{|W(t)|, t\in I\}$ can be provided, when $I$ is taken from a given countable family of intervals covering the positive half-line.

math.PR↗
arXiv2009

On the Supremum of Some Random Dirichlet Polynomials

We study the supremum of some random Dirichlet polynomials with independent coefficients and obtain sharp upper and lower bounds for supremum expectation thus extending the results from our previous work (see http://arXiv.org/abs/math/0703691). Our approach in proving these results is entirely based on methods of stochastic processes, in particular the metric entropy method.

math.PR↗
arXiv2009

Supremum of Random Dirichlet Polynomials with Sub-multiplicative Coefficients

We study the supremum of random Dirichlet polynomials $D_N(t)=\sum_{n=1}^N\varepsilon_n d(n) n^{- s}$, where $(\varepsilon_n)$ is a sequence of independent Rademacher random variables, and $ d $ is a sub-multiplicative function. The approach is gaussian and entirely based on comparison properties of Gaussian processes, with no use of the metric entropy method.

math.PR↗
arXiv2008

On localization in Kronecker's diophantine theorem

We study the localization problem appearing in Kronecker's diophantine theorem. We introduce a probabilistic approach allowing to extend for general $\Q$-linearly independent sequences a result of Túran concerning the sequence $ (\log p_\ell)$, $p_\ell$ being the $\ell$-th prime.

math.NT↗