Encontrar subestruturas aditivas que revelam um certo grau de organização em certos conjuntos contidos nos números naturais é o foco do estudo da combinatória aditiva. Desta área, resultados como os famosos Teorema de Van der Waerden e o Teorema de Szemerédi se destacam, revelando através de métodos combinatoriais que certas propriedades referentes ao tamanho de subconjuntos de inteiros implicam a existência de progressões aritméticas de tamanho arbitrariamente grande. Em meados de 1970, Furstenberg causou certa comoção no meio matemático ao publicar provas para ambos os teoremas usando métodos e ferramentas da teoria ergódica. Apesar de tal abordagem ter apresentado uma nova e profunda ligação entre as áreas, houve certa crítica pelo fato de não gerar resultados originais e por suas limitações (por exemplo, seus resultados costumam ser de caráter assintótico, sem lidar com limitantes e cotas, amplamente conhecidos pelos métodos combinatórios). Tais críticas foram silenciadas quando Ben Green e Terence Tao, usando tais métodos de teoria ergódica, demonstraram a incrível e bela afirmação de que os primos possuem progressões aritméticas de tamanho arbitrariamente grande, dando uma resposta definitiva para um enunciado conjecturado há muito tempo. Certamente, este foi um grande passo na matemática do século XXI. Deste então, novas abordagens foram amplamente estudadas e analisadas, de modo a aumentar ainda mais nossa compreensão sobre estes impressionantes conceitos. / Finding additive substructures that reveal a certain degree of organization in certain sets contained in the set of the natural numbers is the focus of the study of additive combinatorics. From this area, results such as the famous Van der Waerdens Theorem and Szemerédis Theorem stand out, revealing through combinatorial methods that certain properties concerning the size of subsets of integers imply the existence of arbitrarily long arithmetic progressions. In the mid-1970s Furstenberg caused some commotion in the mathematical world by publishing proofs for both theorems using methods and tools of ergodic theory rather than combinatorial methods. Although this approach had presented a new and deep link between those areas, there was some criticism for the lack of original results and some limitations of this technique (for instance, its results usually have an asymptotic flavour without dealing with bounds widely known by combinatorial methods). Such criticisms were silenced when Ben Green and Terence Tao, using such methods of ergodic theory, demonstrated the incredible and beautiful theorem that the primes have arithmetic progressions of arbitrarily large size, giving a definitive answer to a statement conjectured a long time ago. Certainly, this was a major step for the mathematics of the 21st century. Hence, new approaches have been extensively studied and analyzed in order to further increase our understanding of these impressive concepts.

Réalisé en cotutelle avec l'Université Paris-Diderot. / Nous présentons trois résultats en combinatoire additive, un domaine récent à la croisée de la combinatoire, l'analyse harmonique et la théorie analytique des nombres. Le thème unificateur de notre thèse est la détection de structures additives dans les ensembles arithmétiques à faible densité, avec un intérêt particulier pour les aspects quantitatifs. Notre première contribution est une estimation de densité améliorée pour le problème, initié entre autres par Bourgain, de trouver une longue progression arithmétique dans un ensemble somme triple. Notre deuxième résultat consiste en une généralisation des bornes de Sanders pour le théorème de Roth, du cas d'un ensemble dense dans les entiers à celui d'un ensemble à faible croissance additive dans un groupe abélien arbitraire. Finalement, nous étendons les meilleures bornes quantitatives connues pour le théorème de Roth dans les premiers, à tous les systèmes d'équations linéaires invariants par translation et de complexité un. / We present three results in additive combinatorics, a recent field at the interface of combinatorics, harmonic analysis and analytic number theory. The unifying theme in our thesis is the detection of additive structure in arithmetic sets of low density, with an emphasis on quantitative aspects. Our first contribution is an improved density estimate for the problem, initiated by Bourgain and others, of finding a long arithmetic progression in a triple sumset. Our second result is a generalization of Sanders' bounds for Roth's theorem from the dense setting, to the setting of small doubling in an arbitrary abelian group. Finally, we extend the best known quantitative results for Roth's theorem in the primes, to all translation-invariant systems of equations of complexity one.