Doctor of Philosophy / Department of Mathematics / Charles N. Moore / The main purpose of this thesis is to derive the law of the iterated logarithm for tail sums in various contexts in analysis. The various contexts are sums of Rademacher functions, general dyadic martingales, independent random variables and lacunary trigonometric series. We name the law of the iterated logarithm for tail sums as tail law of the iterated logarithm. We first establish the tail law of the iterated logarithm for sums of Rademacher functions and obtain both upper and lower bound in it. Sum of Rademacher functions is a nicely behaved dyadic martingale. With the ideas from the Rademacher case, we then establish the tail law of the iterated logarithm for general dyadic martingales. We obtain both upper and lower bound in the case of martingales. A lower bound is obtained for the law of the iterated logarithm for tail sums of bounded symmetric independent random variables. Lacunary trigonometric series exhibit many of the properties of partial sums of independent random variables. So we finally obtain a lower bound for the tail law of the iterated logarithm for lacunary trigonometric series introduced by Salem and Zygmund.

In this thesis, we make some contributions at the interface between algebra and graph theory. In Chapter 1, we give an overview of the topics and also the definitions and preliminaries. In Chapter 2, we estimate the number of possible types degree patterns of k-lacunary polynomials of degree t < p which split completely modulo p. The result is based on a rather unusual combination of two techniques: a bound on the number of zeros of lacunary polynomials and a bound on the so-called domination number of a graph. In Chapter 3, we deal with the determinant of bipartite graphs. The nullity of a graph G is the multiplicity of 0 in the spectrum of G. Nullity of a (molecular) graph (e.g., a bipartite graph corresponding to an alternant hydrocarbon) has important applications in quantum chemistry and Huckel molecular orbital (HMO) theory. A famous problem, posed by Collatz and Sinogowitz in 1957, asks to characterize all graphs with positive nullity. Clearly, examining the determinant of a graph is a way to attack this problem. In this Chapter, we show that the determinant of a bipartite graph with at least two perfect matchings and with all cycle lengths divisible by four, is zero. In Chapter 4, we first introduce an application of spectral graph theory in proving trigonometric identities. This is a very simple double counting argument that gives very short proofs for some of these identities (and perhaps the only existed proof in some cases!). In the rest of Chapter 4, using some properties of the well-known Chebyshev polynomials, we prove some theorems that allow us to evaluate the number of spanning trees in join of graphs, Cartesian product of graphs, and nearly regular graphs. In the last section of Chapter 4, we obtain the number of spanning trees in an (r,s)-semiregular graph and its line graph. Note that the same results, as in the last section, were proved by I. Sato using zeta functions. But our proofs are much shorter based on some well-known facts from spectral graph theory. Besides, we do not use zeta functions in our arguments. In Chapter 5, we present the conclusion and also some possible projects.

A lacunary series is a Taylor series with large gaps between its non-zero coefficients. In this thesis we exploit these gaps to obtain results of linear independence of values of lacunary series at integer points. As well, we will study different methods found in Diophantine approximation which we use to study arithmetic properties of values of lacunary series at algebraic points. Among these methods will be Mahler's method and a new approach due to Jean-Paul Bézivin.

In this thesis, we make some contributions at the interface between algebra and graph theory. In Chapter 1, we give an overview of the topics and also the definitions and preliminaries. In Chapter 2, we estimate the number of possible types degree patterns of k-lacunary polynomials of degree t < p which split completely modulo p. The result is based on a rather unusual combination of two techniques: a bound on the number of zeros of lacunary polynomials and a bound on the so-called domination number of a graph. In Chapter 3, we deal with the determinant of bipartite graphs. The nullity of a graph G is the multiplicity of 0 in the spectrum of G. Nullity of a (molecular) graph (e.g., a bipartite graph corresponding to an alternant hydrocarbon) has important applications in quantum chemistry and Huckel molecular orbital (HMO) theory. A famous problem, posed by Collatz and Sinogowitz in 1957, asks to characterize all graphs with positive nullity. Clearly, examining the determinant of a graph is a way to attack this problem. In this Chapter, we show that the determinant of a bipartite graph with at least two perfect matchings and with all cycle lengths divisible by four, is zero. In Chapter 4, we first introduce an application of spectral graph theory in proving trigonometric identities. This is a very simple double counting argument that gives very short proofs for some of these identities (and perhaps the only existed proof in some cases!). In the rest of Chapter 4, using some properties of the well-known Chebyshev polynomials, we prove some theorems that allow us to evaluate the number of spanning trees in join of graphs, Cartesian product of graphs, and nearly regular graphs. In the last section of Chapter 4, we obtain the number of spanning trees in an (r,s)-semiregular graph and its line graph. Note that the same results, as in the last section, were proved by I. Sato using zeta functions. But our proofs are much shorter based on some well-known facts from spectral graph theory. Besides, we do not use zeta functions in our arguments. In Chapter 5, we present the conclusion and also some possible projects.

A lacunary series is a Taylor series with large gaps between its non-zero coefficients. In this thesis we exploit these gaps to obtain results of linear independence of values of lacunary series at integer points. As well, we will study different methods found in Diophantine approximation which we use to study arithmetic properties of values of lacunary series at algebraic points. Among these methods will be Mahler's method and a new approach due to Jean-Paul Bézivin.

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El objetivo de esta tesis es desarrollar la interpolación de Birkhoff mediante polinomios lacunarios.En la interpolación algebraica de Birkhoff se determina un polinomio de grado menor que n, para ello se emplean n condiciones que fijan el valor del polinomio o sus derivadas. Los problemas clásicos de interpolación de Lagrange, Taylor, Hermite, Hermite-Sylvester y Abel-Gontcharov son casos particulares de interpolación algebraica de Birkhoff.Un espacio de polinomios lacunarios de dimensión n es el conjunto de los polinomios que pueden generarse por combinación lineal de n potencias distintas de grados, en general, no consecutivos. En particular, cuando tomamos potencias de grados 0,1,.,n-1, se obtiene el espacio de polinomios de grado menor que n, empleado en la interpolación algebraica clásica. En la interpolación algebraica clásica, el número de condiciones determina el espacio de interpolación. En contraste, en la interpolación mediante polinomios lacunarios las condiciones de interpolación determinan únicamente la dimensión del espacio de interpolación y pueden existir una infinidad de espacios sobre los que realizar la interpolación. Esto nos permite construir mejores estrategias de interpolación en ciertos casos, como la interpolación de funciones de gran crecimiento (interpolación de exponenciales y de ramas asintóticas).La aportación de la tesis consiste en la definición de un marco teórico adecuado para la interpolación de Birkhoff mediante polinomios lacunarios y en la extensión al nuevo marco de los principales elementos de la interpolación algebraica de Birkhoff. En concreto, se generaliza la condición de Pólya, se caracteriza la regularidad condicionada, se establecen condiciones suficientes de regularidad ordenada que extienden el teorema de Atkhison-Sharma, se extiende la descomposición normal y se establecen condiciones suficientes de singularidad en los casos indescomponibles.

Pour une suite ⋀ = (λn) satisfaisant la condition de Müntz Σn 1/λn < +∞ et pour p ∈ [1,+∞), on définit l'espace de Müntz Mp⋀ comme le sous-espace fermé de Lp([0, 1]) engendré par les monômes yn : t ↦ tλn. L'espace M∞⋀ est défini de la même façon comme un sous-espace de C([0, 1]). Lorsque la suite (λn + 1/p)n est lacunaire avec un grand indice, nous montrons que la famille (gn) des monômes normalisés dans Lp est (1 + ε)-isométrique à la base canonique de lp. Dans le cas p = +∞, les monômes (yn) forment une famille normalisée et (1 + ε)-isométrique à la base sommante de c. Ces résultats sont un raffinement asymptotique d'un théorème bien connu pour les suites lacunaires. D'autre part, pour p ∈ [1, +∞), nous étudions les mesures de Carleson des espaces de Müntz, c'est-à-dire les mesures boréliennes μ sur [0,1) telles que l'opérateur de plongement Jμ,p : Mp⋀ ⊂ Lp(μ) est borné. Lorsque ⋀ est lacunaire, nous prouvons que si les (gn) sont uniformément bornés dans Lp(μ), alors μ est une mesure de Carleson de Mq⋀ pour tout q > p. Certaines conditionsgéométriques sur μ au voisinage du point 1 sont suffsantes pour garantir la compacité de Jμ,p ou son appartenance à d'autres idéaux d'opérateurs plus fins. Plus précisément, nous estimons les nombres d'approximation de Jμ,p dans le cas lacunaire et nous obtenons même des équivalents pour certaines suites ⋀. Enfin, nous calculons la norme essentielle del'opérateur de moyenne de Cesàro Γp : Lp → Lp : elle est égale à sa norme, c'est-à-dire à p'. Ce résultat est aussi valide pour l'opérateur de Cesàro discret. Nous introduisons les sous-espaces de Müntz des espaces de Cesàro Cesp pour p ∈ [1, +∞]. Nous montrons que la norme essentielle de l'opérateur de multiplication par Ψ est égale à ∥Ψ∥∞ dans l'espace deCesàro, et à |Ψ(1)| dans les espaces de Müntz-Cesàro. / For a sequence ⋀ = (λn) satisfying the Müntz condition Σn 1/λn < +∞ and for p ∈ [1,+∞), we define the Müntz space Mp⋀ as the closed subspace of Lp([0, 1]) spanned by the monomials yn : t ↦ tλn. The space M∞⋀ is defined in the same way as a subspace of C([0, 1]). When the sequence (λn + 1/p)n is lacunary with a large ratio, we prove that the sequence of normalized Müntz monomials (gn) in Lp is (1 + ε)-isometric to the canonical basis of lp. In the case p = +∞, the monomials (yn) form a sequence which is (1 + ε)-isometric to the summing basis of c. These results are asymptotic refinements of a well known theorem for the lacunary sequences. On the other hand, for p ∈ [1, +∞), we investigate the Carleson measures for Müntz spaces, which are defined as the Borel measures μ on [0; 1) such that the embedding operator Jμ,p : Mp⋀ ⊂ Lp(μ) is bounded. When ⋀ is lacunary, we prove that if the (gn) are uniformly bounded in Lp(μ), then for any q > p, the measure μ is a Carleson measure for Mq⋀. These questions are closely related to the behaviour of μ in the neighborhood of 1. Wealso find some geometric conditions about the behaviour of μ near the point 1 that ensure the compactness of Jμ,p, or its membership to some thiner operator ideals. More precisely, we estimate the approximation numbers of Jμ,p in the lacunary case and we even obtain some equivalents for particular lacunary sequences ⋀. At last, we show that the essentialnorm of the Cesàro-mean operator Γp : Lp → Lp coincides with its norm, which is p'. This result is also valid for the Cesàro sequence operator. We introduce some Müntz subspaces of the Cesàro function spaces Cesp, for p ∈ [1, +∞]. We show that the value of the essential norm of the multiplication operator TΨ is ∥Ψ∥∞ in the Cesàaro spaces. In the Müntz-Cesàrospaces, the essential norm of TΨ is equal to |Ψ(1)|.

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