The ring of polynomials over a finite field has many arithmetic properties similar to those of the ring of rational integers. In this thesis, we apply the Hardy-Littlewood circle method to investigate the density of rational points on certain algebraic varieties in function fields. The aim is to establish asymptotic relations that are relatively robust to changes in the characteristic of the base finite field. More notably, in the case when the characteristic is "small", the results are sharper than their integer analogues.

The ring of polynomials over a finite field has many arithmetic properties similar to those of the ring of rational integers. In this thesis, we apply the Hardy-Littlewood circle method to investigate the density of rational points on certain algebraic varieties in function fields. The aim is to establish asymptotic relations that are relatively robust to changes in the characteristic of the base finite field. More notably, in the case when the characteristic is "small", the results are sharper than their integer analogues.

The self-power map is the function from the set of natural numbers to itself which sends the number $n$ to $n^n$. Motivated by applications to cryptography, we consider the image of this map modulo a prime $p$. We study the question of how large $x$ must be so that $n^n \equiv a \bmod p$ has a solution with $1 \le n \le x$, for every residue class $a$ modulo $p$. While $n^n \bmod p$ is not uniformly distributed, it does appear to behave in certain ways as a random function. We give a heuristic argument to show that the expected $x$ is approximately ${p^2\log \phi(p-1)/\phi(p-1)}$, using the coupon collector problem as a model. Rigorously, we prove the bound $x <p^{2-\alpha}$ for sufficiently large $p$ and a fixed constant $\alpha > 0$ independent of $p$, using a counting argument and exponential sum bounds. Additionally, we prove nontrivial bounds on the number of solutions of $n^n \equiv a \bmod p$ for a fixed residue class $a$ when $1 \le n \le x$, extending the known bounds when $1 \le n \le p-1$.

The self-power map is the function from the set of natural numbers to itself which sends the number $n$ to $n^n$. Motivated by applications to cryptography, we consider the image of this map modulo a prime $p$. We study the question of how large $x$ must be so that $n^n \equiv a \bmod p$ has a solution with $1 \le n \le x$, for every residue class $a$ modulo $p$. While $n^n \bmod p$ is not uniformly distributed, it does appear to behave in certain ways as a random function. We give a heuristic argument to show that the expected $x$ is approximately ${p^2\log \phi(p-1)/\phi(p-1)}$, using the coupon collector problem as a model. Rigorously, we prove the bound $x <p^{2-\alpha}$ for sufficiently large $p$ and a fixed constant $\alpha > 0$ independent of $p$, using a counting argument and exponential sum bounds. Additionally, we prove nontrivial bounds on the number of solutions of $n^n \equiv a \bmod p$ for a fixed residue class $a$ when $1 \le n \le x$, extending the known bounds when $1 \le n \le p-1$.

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La présente thèse a été fortement influencée par deux conjectures, l'une de Gelfond et l'autre de Sarnak.En 1968, Gelfond a prouvé que la somme des chiffres modulo m est asymtotiquement équirépartie dans des progressions arithmétiques, et il a formulé trois problèmes nouveaux.Le deuxième et le troisième problèmes traitent des sommes des chiffres pour les nombres premiers et les suites polynomiales.En ce qui concerne les nombres premiers et les carrés, Mauduit et Rivat ont résolu ces problèmes en 2010 et 2009, respectivement.Drmota, Mauduit et Rivat ont réussi généraliser le résultat concernant la suite des sommes des chiffres des carrés.Ils ont démontré que chaque bloc apparaît asymptotiquement avec la même fréquence.Selon la conjecture de Sarnak, il n'y a pas de corrélation entre la fonction de Möbius et des fonctions simples.La présente thèse traite de la répartition de suites automatiques le long de sous-suites particulières ainsi que d'autres propriétés de suites automatiques.Selon l'un des résultats principaux du présent travail, toutes les suites automatiques vérifient la conjecture de Sarnak.Moyennant une approche légèrement modifiée, nous traitons également la répartition de suites automatiques le long de la suite des nombres premiers.Dans le cadre du traitement de suites automatiques générales, nous avons mis au point une nouvelle structure destinée aux automates finisdéterministes ouvrant une vision nouvelle pour les automates et/ou les suites automatiques.Nous étendons les résultat de Drmota, Mauduit et Rivat concernant les suites digitales.Cette approche peut également être considérée comme une généralisation du troisième problème de Gelfond. / The present dissertation was inspired by two conjectures, one by Gelfond and one of Sarnak.In 1968 Gelfond proved that the sum of digits modulo m is asymptotically equally distributed along arithmetic progressions.Furthermore, he stated three problems which are nowadays called Gelfond problems.The second and third questions are concerned with the sum of digits of prime numbers and polynomial subsequences.Mauduit and Rivat were able to solve these problems for primes and squares in 2010 and 2009 respectively.Drmota, Mauduit and Rivat generalized the result concerning the sequence of the sum of digits of squares.They showed that each block appears asymptotically equally frequently.Sarnak conjectured in 2010 that the Mobius function does not correlate with deterministic functions.This dissertation deals with the distribution of automatic sequences along special subsequences and other properties of automatic sequences.A main result of this thesis is that all automatic sequences satisfy the Sarnak conjecture.Through a slightly modified approach, we also deal with the distribution of automatic sequences along the subsequence of primes.In the course of the treatment of general automatic sequences, a new structure for deterministic finite automata is developed,which allows a new view for automata or automatic sequences.We extend the result of Drmota, Mauduit and Rivat to digital sequences.This is also a generalization of the third Gelfond problem.

Dans ce travail nous étudions les propriétés arithmétiques et statistiques d'une nouvelle classe de fonctions de comptage des chiffres appelées fonctions digitales restreintes. Nous présentons tout d'abord les principales propriétés des suites engendrées par une substitution ou un $q$-automate ainsi que la suite célèbre de Thue-Morse et ses généralisations, puis nous comparons ces notions avec celle de fonction digitale restreinte.Nous étudions ensuite les sommes d'exponentielles associées à ces fonctions digitales restreintes ainsi que leur application d'une part à l'étude de la répartition modulo 1 des fonctions digitales restreintes et d'autre part à l'étude des propriétés statistiques des suites arithmétiques définies par des fonctions digitales restreintes.Dans la dernière partie de ce travail on étudie la représentation géométrique de ces sommes d'exponentielle à la lumière des travaux antérieurs de Dekking et Mendès-France ce qui nous conduit à énoncer plusieurs problèmes ouverts. / In this work we study the arithmetic and statistic properties of a new class of digital counting functions called restricted digital functions. We first present the main properties of sequences generated by a substitution or a $q$-automate followed by presenting the famous Thue-Morse sequence and its generalizations, then we compare these notions with the one of the restricted digital function.We then study the exponential sums associated with these restricted digital function and their implementation on the one hand to the study of uniform distribution modulo 1 of these restricted digital functions and on the other, to the study of the statistical properties of the arithmetic sequences defined by restricted digital functions.In the last part of this work we study the geometric representation of these exponential sums in the light of previous works of Dekking and Mendès-France which leads us to announce several open problems.