Van Der Corput's Lemma in Number Theory and Analysis and its Applications to Abelian Varieties with Prescribed Groups

Let A be an Abelian variety over a finite field Fq. We are interested in knowing the distribution of the groups A(Fq) of rational points on A as we run over all varieties defined over Fq. In particular, we want to show that they are in general not too "split". For the case of dimension 1 (elliptic curves) and dimension 2 (Abelian surfaces), there are some theoretical results due to David and her collaborators, but the general case is open. We are interested in Abelian Varieties of dimension 3. We use Rybakov's criterion, which relates the existence of a given abstract group as the group of points of some Abelian variety to properties of the characteristic polynomial of the variety. We can use it to derive precise properties and then we use the fact that some sequence of monomials of five variables is uniformly distributed modulo one to obtain stronger results that will hold with probability one. By Weyl's criterion, equidistribution follows by bounding exponential sums, and in order to do so, we will use a combination of different methods. We are particularly interested in Van Der Corput's lemma. It has a continuous version that exhibits the decay of oscillatory integrals and a discrete version that gives a bound for exponential sums. We will see the relation between these two versions and how they apply to the original problem of Abelian varieties

The density of integral points on complete intersections

In this paper, an upper bound for the number of integral points of bounded height on an affine complete intersection defined over $\mathbb{Z}$ is proven. The proof uses an extension to complete intersections of the method used for hypersurfaces by Heath-Brown, the so called q-analogue of van der Corput's AB process.Comment: 24 pages, Appendix by Per Salberger; typos correcte

Dispersive stability of infinite dimensional Hamiltonian systems on lattices

We derive dispersive stability results for oscillator chains like the FPU chain or the discrete Klein-Gordon chain. If the nonlinearity is of degree higher than 4, then small localized initial data decay like in the linear case. For this, we provide sharp decay estimates for the linearized problem using oscillatory integrals and avoiding the nonoptimal interpolation between different $ell^p$ space

A study of correlation of sequences.

by Wai Ho Mow.Thesis (Ph.D.)--Chinese University of Hong Kong, 1993.Includes bibliographical references (leaves 116-124).Chapter 1 --- Introduction --- p.1Chapter 1.1 --- Spread Spectrum Technique --- p.2Chapter 1.1.1 --- Pulse Compression Radars --- p.3Chapter 1.1.2 --- Spread Spectrum Multiple Access Systems --- p.6Chapter 1.2 --- Definitions and Notations --- p.8Chapter 1.3 --- Organization of this Thesis --- p.12Chapter 2 --- Lower Bounds on Correlation of Sequences --- p.15Chapter 2.1 --- Welch's Lower Bounds and Sarwate's Generalization --- p.16Chapter 2.2 --- A New Construction and Bounds on Odd Correlation --- p.23Chapter 2.3 --- Known Sequence Sets Touching the Correlation Bounds --- p.26Chapter 2.4 --- Remarks on Other Bounds --- p.27Chapter 3 --- Perfect Polyphase Sequences: A Unified Approach --- p.29Chapter 3.1 --- Generalized Bent Functions and Perfect Polyphase Sequences --- p.30Chapter 3.2 --- The General Construction of Chung and Kumar --- p.32Chapter 3.3 --- Classification of Known Constructions ...........； --- p.34Chapter 3.4 --- A Unified Construction --- p.39Chapter 3.5 --- Desired Properties of Sequences --- p.41Chapter 3.6 --- Proof of the Main Theorem --- p.45Chapter 3.7 --- Counting the Number of Perfect Polyphase Sequences --- p.49Chapter 3.8 --- Results of Exhaustive Searches --- p.53Chapter 3.9 --- A New Conjecture and Its Implications --- p.55Chapter 3.10 --- Sets of Perfect Polyphase Sequences --- p.58Chapter 4 --- Aperiodic Autocorrelation of Generalized P3/P4 Codes --- p.61Chapter 4.1 --- Some Famous Polyphase Pulse Compression Codes --- p.62Chapter 4.2 --- Generalized P3/P4 Codes --- p.65Chapter 4.3 --- Asymptotic Peak-to-Side-Peak Ratio --- p.66Chapter 4.4 --- Lower Bounds on Peak-to-Side-Peak Ratio --- p.67Chapter 4.5 --- Even-Odd Transformation and Phase Alphabet --- p.70Chapter 5 --- Upper Bounds on Partial Exponential Sums --- p.77Chapter 5.1 --- Gauss-like Exponential Sums --- p.77Chapter 5.1.1 --- Background --- p.79Chapter 5.1.2 --- Symmetry of gL(m) and hL(m) --- p.80Chapter 5.1.3 --- Characterization on the First Quarter of gL(m) --- p.83Chapter 5.1.4 --- Characterization on the First Quarter of hL(m) --- p.90Chapter 5.1.5 --- Bounds on the Diameters of GL(m) and HL(m) --- p.94Chapter 5.2 --- More General Exponential Sums --- p.98Chapter 5.2.1 --- A Result of van der Corput --- p.99Chapter 6 --- McEliece's Open Problem on Minimax Aperiodic Correlation --- p.102Chapter 6.1 --- Statement of the Problem --- p.102Chapter 6.2 --- A Set of Two Sequences --- p.105Chapter 6.3 --- A Set of K Sequences --- p.110Chapter 7 --- Conclusion --- p.113Bibliography --- p.12

Local well-posedness of the higher order nonlinear Schr\"odinger equation on the half-line: single boundary condition case

We establish local well-posedness for the higher-order nonlinear Schr\"odinger equation, formulated on the half-line. We consider the scenario of associated coefficients such that only one boundary condition is required, which is assumed to be Dirichlet type. Our functional framework centers around fractional Sobolev spaces. We treat both high regularity and low regularity solutions: in the former setting, the relevant nonlinearity can be handled via the Banach algebra property; in the latter setting, however, delicate Strichartz estimates must be established. This task is especially challenging in the framework of nonhomogeneous initial-boundary value problems, as it involves proving boundary-type Strichartz estimates that are not common in the study of initial value problems. The linear analysis, which is the core of this work, crucially relies on a weak solution formulation defined through the novel solution formulae obtained via the Fokas method. In this connection, we note that the higher-order Schr\"odinger equation comes with an increased level of difficulty due to the presence of more than one spatial derivative. This feature manifests itself via several complications throughout the analysis, including (i) analyticity issues related to complex square roots, which require careful treatment of branch cuts and deformations of integration contours; (ii) singularities that emerge upon changes of variables in the Fourier analysis arguments; (iii) complicated oscillatory kernels in the weak solution formula for the linear initial-boundary value problem, which require a subtle analysis of the dispersion in terms of the regularity of the boundary data. The present work provides a first, complete treatment via the Fokas method of a nonhomogeneous initial-boundary value problem for a partial differential equation associated with a multi-term linear differential operator.Comment: 30 pages, 2 figure

Diophantine approximation and prime numbers

In the first part of this thesis various problems in diophantine approximation are considered, which generalize well known theorems of Dirichlet and Kronecker. A brief survey is presented in the first chapter, including a discussion on the scope of elementary methods. It is demonstrated here that stronger results are possible by elementary means than have previously been obtained. In the subsequent chapters non-elementary methods are used. Results are proved for fractional parts of quadratic forms in several variables which improve upon previous work. New theorems are demonstrated for the distribution modulo one of 'almost all" additive forms in many variables, including the particularly interesting case of a linear form in positive variables. In chapter four new bounds are given for exponential sums over primes, which greatly improve upon the work of I.M. Vinogradov. Some applications to diophantine approximation problems involving primes are given in chapters 4 and 5, the latter chapter also improving upon previous work on the problem of a linear form in three prime variables. In the second section, topics in multiplicative number theory are discussed. It is shown that almost-primes are very well distributed in almost all very short intervals, improving upon previous work by a considerable factor. Sieve methods are then employed to tackle three other problems. New results are in this way obtained for primes in short intervals, for the distribution of the square roots of primes (modulo one), and for the distribution of [alpha] modulo one for irrational [alpha]. This last chapter contains a new method for tackling sums over primes which has other applications.<p

Topics in analytic number theory

Le présent document est un compte-rendu de quatre présentations que j'ai faites au congrès de Théorie des Nombres Québec-Maine entre 2013 et 2016. Au fil des ans, j'ai effectué quelques améliorations et corrections aux documents originaux. Le contenu, l'esprit et l'organisation sont restés essentiellement inchangés. Les quatre sujets sont fondamentalement distincts tout en étant dans un même cercle d'idées. Le premier chapitre traite d'un certain nombre de sujets en relation avec le comportement moyen de certaines fonctions multiplicatives, dans un ensemble bien précis, qui partagent plusieurs propriétés avec la fonction indicatrice des nombres libres de puissance k-ième. En particulier, on y établit plusieurs estimations de la variance dans des intervalles courts et dans des progressions arithmétiques. Le deuxième chapitre étudie un problème du crible combinatoire. Il y est question d'établir une majoration analogue à la célèbre inégalité de Brun-Titchmarsh, mais pour les nombres libres de puissance k-ième. Après quelques remarques élémentaires, on établit une nouvelle inégalité en supposant une conjecture forte en lien avec la densité maximale d'une suite de nombres ayant un diviseur de la forme pk 1pk 2 où p1 et p2 sont des nombres premiers qui satisfont certaines conditions. La méthode fournit aussi une majoration effective pour le nombre de nombres libres de puissance k-ième dans un intervalle [x + 1, x + h] lorsque h est petit par rapport à x. Le troisième chapitre, écrit en collaboration avec Jean-Marie De Koninck, établit des inégalités particulières pour la fonction τ(n) qui compte le nombre de diviseurs de n. L'objectif est d'obtenir une majoration de τ(n) qui ne dépend pas des facteurs premiers de n, mais seulement du nombre de facteurs premiers distincts de n et de son ordre de grandeur, i.e. de log n. L'inégalité principale (Théorèmes 3.4 et 3.5) a nécessité un bon volume de calcul sur ordinateur, et donc beaucoup de programmation avec Maple. Finalement, le Chapitre 4 est le début d'une étude du nombre de points entiers près d'une courbe dans l'espace R3. Le problème peut aussi être vu comme celui du nombre de points entiers près de deux courbes dans le plan Euclidien simultanément. L'objectif principal est d'utiliser l'information des deux courbes de façon nontriviale, soit de faire mieux que les meilleurs résultats connus pour une seule courbe. Étant donné la complexité du problème déjà en deux dimensions et du nombre de méthodes disponibles, il nous a semblé impossible de faire un traitement complet de la question. On s'est donc concentré sur une méthode qui utilise des approximations linéaires. Cette dernière peut sans doute être substantiellement améliorée.Résumé en anglaisThéorie analytique des nombre