This dissertation consists of three parts. In the first part we consider the equidistribution of roots of quadratic congruences. The roots of quadratic congruences are known to be equidistributed. However,we establish a bound for the discrepancy of this sequence using a spectral method involvingautomorphic forms, especially Kuznetsov's formula, together with an Erdős-Turán inequality. Then we discuss the implications of our discrepancy estimate for the reducibility problem of arctangents of integers. In the second and third part of this dissertation we consider some aspects of Farey fractions. The set of Farey fractions of order at most [mathematical formula] is, of course, a classical object in Analytic Number Theory. Our interest here is in certain sumsets of Farey fractions. Also, in this dissertation we study Farey fractions by working in the quotient group Q/Z, which is the modern point of view. We first derive an identity which involves the structure of Farey fractions in the group ring of Q/Z. Then we use these identities to estimate the asymptotic magnitude of the size of the sumset [mathematical formula]. Our method uses results about divisors in short intervals due to K. Ford. We also prove a new form of the Erdős-Turán inequality in which the usual complex exponential functions are replaced by a special family of functions which are orthogonal in L²(R/Z). / text

Thesis advisor: Dubi Kelmer / We compute effective error rates for the equidistribution of translates of diagonal orbits on Hilbert modular varieties. The translation is determined by n real parameters and our results require the assumption that all parameters are non-zero. The error rate is given in explicit polynomial terms of the translation parameters and Sobolev type norms of the test functions. The effective equidistribution is applied to give counting estimates for binary quadratic forms of square discriminant over real number rings. / Thesis (PhD) — Boston College, 2022. / Submitted to: Boston College. Graduate School of Arts and Sciences. / Discipline: Mathematics.

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Nous étudions la propriété RD en terme de décroissance de coefficients matriciels de représentations unitaires. Nous nous concentrons en particulier sur des représentations provenant de l'action des groupes de Lie et de groupes discrets sur un "bord" approprié. Ces actions produisent des rerésentations unitaires à normalisation prés. Nous utilisons des techniques d'analyse harmonique et de théorie ergodique pour amorcer une nouvelle approche de la conjecture de Valette. / We study property RD in terms of decay of matrix coefficients for unitary representations. We focus our attention on unitary representations arising from action of Lie groups and discrete groups of isometries of a CAT(-1) space on their appropriate boundary. We use some techniques of harmonic analysis, and ergodic theory to start a new approach of Valette's conjecture

京都大学 / 新制・課程博士 / 博士(理学) / 甲第24387号 / 理博第4886号 / 新制||理||1699(附属図書館) / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)教授 森脇 淳, 教授 雪江 明彦, 教授 吉川 謙一 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM

Dans ce mémoire, nous nous intéressons aux surfaces de translation. Ce sont des surfaces compactes munies d'une métrique plate, qui possèdent des singularités coniques et sur lesquelles, on peut choisir une direction verticale. De manière équivalente, une surface de translation est aussi une 1-forme holomorphe sur une surface de Riemann. Des exemples majeurs de telles surfaces sont les surfaces obtenues par “ dépliage ” de billards rationnels.Nous identifions deux surfaces de translation images l'une de l'autre par une isométrie préservant l'orientation et la direction verticale. La classe d'une surface par cette relation d'équivalence est encore une surface de translation que l'on appelle surface réduite de la surface de départ.Nous définissons les difféomorphismes affines d'une surface de translation comme les difféomorphismes de cette surface dont la différentielle est constante. Ils forment un groupe appelé le groupe affine de la surface.Le groupe SL(2,IR) agit linéairement sur l'ensemble des surfaces de translation. Le stabilisateur de la surface réduite d'une surface de translation est appelé le groupe de Veech de la surface de translation. Les éléments du groupe de Veech sont en fait les matrices jacobiennes des difféomorphismes affines. Ce groupe est un outil indispensable dans l'étude des surfaces de translation et notre travail en est une illustration. Si le groupe de Veech est un réseau de SL(2,IR), la surface est appelée surface de Veech.L'objectif de ce mémoire est de démontrer que, sur une surface de Veech donnée, les orbites denses du groupe affine s'équirépartissent sur la surface. Nous précisons bien sûr la notion d'équirépartition utilisée. Il est important de noter que les orbites qui ne sont pas denses sont finies et qu'il y en a au plus un nombre dénombrable. Ce résultat est d'abord établi pour la surface réduite de la surface de translation et permet d'en déduire le théorème pour la surface de départ. / In this thesis, we study translation surfaces. These are compact surfaces equipped with a flat metric and conical singularities. A vertical direction is fixed. Translation surfaces are in one to one correspondence with holomorphic 1-forms on Riemann surfaces. Important examples of translation surfaces arise from unfolding billiards in rational polygons.Two translation surfaces are identified if they are obtained one from the other by an isometry preserving the orientation and the vertical direction. The equivalence class of a surface is still a translation surface called the reduced surface. Affine diffeomorphisms on a translation surface are diffeomorphisms whose differential is constant. They form a group called the affine group. The group SL(2,R) acts linearly on the set of translation surfaces. The stabilizer of the reduced surface is the Veech group of the translation surface. The elements of the Veech group are in fact the derivative of the affine diffeomorphisms. This group is of great importance in the study of translation surfaces and our work illustrate this phenomenon. If the Veech group is a lattice in SL(2,R), the surface is called a Veech surface. The goal of this thesis is to prove that dense orbit of the affine group on a Veech surface are equidistributed in the surface. One has to explain precisely what equidistribution means in this context. It is important to notice that non dense orbits are finite and that the number of these orbits is at most countable. The result is first of all established for reduced surfaces and we deduce a general result for all surfaces.

Submitted by Cássia Santos (cassia.bcufg@gmail.com) on 2017-07-05T14:04:12Z No. of bitstreams: 2 Dissertação - Jefferson Marques Santos - 2017.pdf: 1510253 bytes, checksum: fa6dbf92bac6614d3ce705a47bbe41b8 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Approved for entry into archive by Luciana Ferreira (lucgeral@gmail.com) on 2017-07-10T14:31:22Z (GMT) No. of bitstreams: 2 Dissertação - Jefferson Marques Santos - 2017.pdf: 1510253 bytes, checksum: fa6dbf92bac6614d3ce705a47bbe41b8 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Made available in DSpace on 2017-07-10T14:31:23Z (GMT). No. of bitstreams: 2 Dissertação - Jefferson Marques Santos - 2017.pdf: 1510253 bytes, checksum: fa6dbf92bac6614d3ce705a47bbe41b8 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Previous issue date: 2017-06-20 / The concept of roots of a polynomial is quite simple but has several applications. This concept extends more generally to the case of "small" algebraic points sequences in a curve. This dissertation aims to estimate the size of algebraic numbers by means of Weil height. In addition to showing that they are distributed evenly around the unit circle, through Bilu Equidistribution Theorem. / O conceito de raízes de um polinômio é bastante simples mas possui várias aplicações. Este conceito se estende de forma mais geral para o caso de sequências de pontos algébricos “pequenos” em uma curva. Esta dissertação tem por objetivo estimar o tamanho de números algébricos por meio da altura de Weil. Além de mostrar que os mesmos se distribuem uniformemente em torno do círculo unitário, por meio do Teorema de Equidistribuição de Bilu.

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This PhD thesis consists of a summary and five papers which all deal with asymptotic problems on certain homogeneous spaces. In Paper I we prove asymptotic equidistribution results for pieces of large closed horospheres in cofinite hyperbolic manifolds of arbitrary dimension. All our results are given with precise estimates on the rates of convergence to equidistribution. Papers II and III are concerned with statistical problems on the space of n-dimensional lattices of covolume one. In Paper II we study the distribution of lengths of non-zero lattice vectors in a random lattice of large dimension. We prove that these lengths, when properly normalized, determine a stochastic process that, as the dimension n tends to infinity, converges weakly to a Poisson process on the positive real line with intensity 1/2. In Paper III we complement this result by proving that the asymptotic distribution of the angles between the shortest non-zero vectors in a random lattice is that of a family of independent Gaussians. In Papers IV and V we investigate the value distribution of the Epstein zeta function along the real axis. In Paper IV we determine the asymptotic value distribution and moments of the Epstein zeta function to the right of the critical strip as the dimension of the underlying space of lattices tends to infinity. In Paper V we determine the asymptotic value distribution of the Epstein zeta function also in the critical strip. As a special case we deduce a result on the asymptotic value distribution of the height function for flat tori. Furthermore, applying our results we discuss a question posed by Sarnak and Strömbergsson as to whether there in large dimensions exist lattices for which the Epstein zeta function has no zeros on the positive real line.

L'objet de cette thèse est l'étude de certaines propriétés arithmétiques et combinatoires de la fonction somme des chiffres. Nous commençons par étudier les sommes d'exponentielles de la forme $dissum_{nleq x}expleft(2ipileft(frac{l}{m}S_q(n)+frac{k}{m'}S_{q}(n+1)+theta nright)right)$ en vue de montrer un résultat d'équirépartition modulo $1$ et un théorème probabiliste d'ErdH{o}s-Kac. Ensuite, on va généraliser un problème dû à Gelfond concernant l'étude de la répartition dans les progressions arithmétiques de la fonction somme des chiffres au cas des nombres ellipséphiques. En particulier, on donne un théorème analogue à celui d'Erdös, Mauduit et S'arközy sur l'uniforme répartition des entiers ellipséphiques dans les progressions arithmétiques sous une contrainte sur la somme des chiffres. Enfin, une étude de l'ordre moyen de certaines fonctions arithmétiques soumises à des contraintes digitales est faite en conséquence des travaux de Mkaouar et Wannès. / The aim of this thesis is the study of some arithmetic and combinatoric properties of the sum of digits function. We start by the study of exponential sums of the form $dissum_{nleq x}expleft(2ipileft(frac{l}{m}S_q(n)+frac{k}{m'}S_q(n+1)+theta nright)right)$ in order to establish a result of equidistribution modulo $1$ in addition to a probabilistic theorem of the kind ErdH{o}s-Kac. Then, we generalize a problem due to Gelfond concerning the distribution in residue classes of the sum of digits function in the case of integers with missing digits. Besides, we give a similar result to that of ErdH{o}s, Mauduit and S'ark"{o}zy on the uniform distribution of integers with missing digits in arithmetic progressions under a constraint on the sum of digits. Finally, a study of the order of magnitude of some arithmetical functions under digital constraints is done as a consequence of the works of Mkaouar and Wannès.