On Near-Linear-Time Algorithms for Dense Subset Sum

In the Subset Sum problem we are given a set of $n$ positive integers $X$ and a target $t$ and are asked whether some subset of $X$ sums to $t$. Natural parameters for this problem that have been studied in the literature are $n$ and $t$ as well as the maximum input number $\rm{mx}_X$ and the sum of all input numbers $\Sigma_X$. In this paper we study the dense case of Subset Sum, where all these parameters are polynomial in $n$. In this regime, standard pseudo-polynomial algorithms solve Subset Sum in polynomial time $n^{O(1)}$. Our main question is: When can dense Subset Sum be solved in near-linear time $\tilde{O}(n)$? We provide an essentially complete dichotomy by designing improved algorithms and proving conditional lower bounds, thereby determining essentially all settings of the parameters $n,t,\rm{mx}_X,\Sigma_X$ for which dense Subset Sum is in time $\tilde{O}(n)$. For notational convenience we assume without loss of generality that $t \ge \rm{mx}_X$ (as larger numbers can be ignored) and $t \le \Sigma_X/2$ (using symmetry). Then our dichotomy reads as follows: - By reviving and improving an additive-combinatorics-based approach by Galil and Margalit [SICOMP'91], we show that Subset Sum is in near-linear time $\tilde{O}(n)$ if $t \gg \rm{mx}_X \Sigma_X/n^2$. - We prove a matching conditional lower bound: If Subset Sum is in near-linear time for any setting with $t \ll \rm{mx}_X \Sigma_X/n^2$, then the Strong Exponential Time Hypothesis and the Strong k-Sum Hypothesis fail. We also generalize our algorithm from sets to multi-sets, albeit with non-matching upper and lower bounds

The elliptic sieve and Brauer groups

A theorem of Serre states that almost all plane conics over Q ${{\mathbb {Q}}}$ have no rational point. We prove an analogue of this for families of conics parametrised by elliptic curves using elliptic divisibility sequences and a version of the Selberg sieve for elliptic curves. We also give more general results for specialisations of Brauer groups, which yields applications to norm form equations

Extremal Results in and out of Additive Combinatorics

In this thesis, we study several related topics in extremal combinatorics, all tied together by various themes from additive combinatorics and combinatorial geometry. First, we discuss some extremal problems where local properties are used to derive global properties. That is, we consider a given configuration where every small piece of the configuration satisfies some restriction, and use this local property to derive global properties of the entire configuration. We study one such Ramsey problem of Erdős and Shelah, where the configurations are complete graphs with colored edges and every small induced subgraph contains many distinct colors. Our bounds for this Ramsey problem show that the known probabilistic construction is tight in various cases. We study one discrete geometry variant, also by Erdős, where we have a set of points in the plane such that every small subset spans many distinct distances. Finally, we consider an arithmetic variant, suggested by Dvir, where we are given sets of real numbers such that every small subset has a large difference set. In Chapter 2, we derive new bounds for all of the above problems. Along the way, we also essentially answer a question of Erdős and Gyárfás. Second, we study the behavior of expanding polynomials on sets with additive or multiplicative structure. Given an arbitrary set of real numbers A and a two-variate polynomial f with real coefficients, a remarkable theorem of Elekes and Rónyai from 2000 states that the size |f(A,A)| of the image of f on the cartesian product A × A grows asymptotically faster than |A|, unless f exhibits additive or multiplicative structure. Finding the best quantitative bounds for this intriguing phenomenon (and for variants of it) has generated a lot of interest over the years due to its intimate connection with the sum-product problem in additive combinatorics. In Chapter 3, we discuss new bounds for |f(A,A)| when the set A has few sums or few products. Another central problem in additive combinatorics is the problem of finding good quantitative bounds for maximal progression-free sets in the integers (or various other groups). In 2017, a major breakthrough of Croot, Lev and Pach took the community by surprise with impressive new bounds for the problem in ℤ4n and in higher order 2-abelian groups. Their new polynomial method was quickly adapted by Ellenberg and Gijswijt to show a similar strong result for the size of the largest three-term progression free subset of &#x1D53D;qn where q is an odd prime power, the so-called cap set problem. This new set of ideas has subsequently led to very exciting developments in a vast range of topics. The rest of the thesis will be dedicated to discussing my joint results around these new developents. In Chapter 4, we develop a new multi-layered polynomial method approach to derive improved bounds for the largest three-term progression free set in ℤ8n (which also improve on the Croot-Lev-Pach bounds for a large family of higher order 2-abelian groups). In Chapter 5, we generalize the Ellenberg-Gijswijt bound for the cap set problem to random progression-free subsets of &#x1D53D;qn, improving on a theorem of Tao and Vu. A result of this type enables one to find four term progressions-free sets which contain three-term progressions in all of their large subsets (with good quantitative bounds), but which do not contain too many three-term progressions overall. Motivated by this application, in Chapter 6 we continue this investigation and study further the question of determining the maximum total number of 3APs in a given 4AP-free set. We show in general, for all fixed integers k &gt; s ≥ 3, that if fs,k(n) denotes the maximum possible number of s-term arithmetic progressions in a set of n integers which contains no k-term arithmetic progression, then fs,k(n) = n2-o(1). This answers an old question of Erdős. In Chapter 7, we study some limitations of the Croot-Lev-Pach approach and discuss some problems at the intersection of extremal set theory and combinatorial geometry where one can use additional linear algebraic ideas to go slightly beyond the Croot-Lev-Pach method.</p

Multiplicative functions with small partial sums and an estimate of Linnik revisited

Cette thèse se compose de deux projets. Le premier concerne la structure des fonctions multiplicatives dont les moyennes sont petites. En particulier, dans ce projet, nous établissons le comportement moyen des valeurs $f(p)$ de $f$ aux nombres premiers pour des fonctions $f$ multiplicatives appropriées lorsque leurs sommes partielles $\sum_{n\leqslant x}f(n)$ sont plus petites que leur borne supérieure triviale par un facteur d′une puissance de $\log x$. Ce résultat poursuit un travail antérieur de Koukoulopoulos et Soundararajan et il est construit sur des idées provenant du traitement plus soigné de Koukoulopoulos sur le cas special des fonctions multiplicatives bornées. Le deuxième projet de la thèse est inspiré par un analogue d’une estimation que Linnik a déduit dans sa tentative de prouver son célèbre théorème concernant la taille du plus petit nombre premier d’une progression arithmétique. Cette estimation fournit une formule asymptotique fortement uniforme pour les sommes de la fonction de von Mangoldt $\Lambda$ sur les progressions arithmétiques. Dans la littérature, ses preuves existantes utilisent des informations non triviales sur les zéros des fonctions $L$ de Dirichlet $L(\cdot,\chi)$ et le but du deuxième projet est de présenter une approche différente, plus élémentaire qui récupère cette estimation en évitant la “langue” de ces zéros. Pour le développement de cette méthode alternative, nous utilisons des idées qui apparaissent dans le grand crible prétentieux (pretentious large sieve) de Granville, Harper et Soundararajan. De plus, comme dans le cas du premier projet, nous empruntons également des idées du travail de Koukoulopoulos sur la structure des fonctions multiplicatives bornées à petites moyennes.This thesis consists of two projects. The first one is concerned with the structure of multiplicative functions whose averages are small. In particular, in this project, we establish the average behaviour of the prime values $f(p)$ for suitable multiplicative functions $f$ when their partial sums $\sum_{n\leqslant x}f(n)$ admit logarithmic cancellations over their trivial upper bound. This result extends previous related work of Koukoulopoulos and Soundararajan and it is built upon ideas coming from the more careful treatment of Koukoulopoulos on the special case of bounded multiplicative functions. The second project of the dissertation is inspired by an analogue of an estimate that Linnik deduced in his attempt to prove his celebrated theorem regarding the size of the smallest prime number of an arithmetic progression. This estimate provides a strongly uniform asymptotic formula for the sums of the von Mangoldt function $\Lambda$ on arithmetic progressions. In the literature, its existing proofs involve non-trivial information about the zeroes of Dirichlet $L$-functions $L(\cdot,\chi)$ and the purpose of the second project is to present a different, more elementary approach which recovers this estimate by avoiding the “language” of those zeroes. For the development of this alternative method, we make use of ideas that appear in the pretentious large sieve of Granville, Harper and Soundararajan. Moreover, as in the case of the first project, we also borrow insights from the work of Koukoulopoulos on the structure of bounded multiplicative functions with small averages

Rough Numbers and Variations on the Erdős--Kac Theorem

The study of arithmetic functions, functions with domain N and codomain C, has been a central topic in number theory. This work is dedicated to the study of the distribution of arithmetic functions of great interest in analytic and probabilistic number theory. In the first part, we study the distribution of positive integers free of prime factors less than or equal to any given real number y\u3e=1. Denoting by Phi(x,y) the count of these numbers up to any given x\u3e=y, we show, by a combination of analytic methods and sieves, that Phi(x,y)\u3c0.6x/\log y holds uniformly for all 3\u3c=y\u3c=sqrt{x}, improving upon an earlier result of the author in the same range. We also prove numerically explicit estimates of the de Bruijn type for Phi(x,y) which are applicable in wide ranges. In the second part, we turn to the topic of weighted Erdős--Kac theorems for general additive functions. Our results concern the distribution of additive functions f(n) weighted by nonnegative multiplicative functions alpha(n) in a wide class. Building on the moment method of Granville, Soundararajan, Khan, Milinovich and Subedi, we establish uniform asymptotic formulas for the moments of f(n) with a suitable growth rate. Our method also enables us to prove a qualitative result on the moments which extends a theorem of Delange and Halberstam on the moments of additive functions. As a consequence, we obtain a weighted analogue of the Kubilius--Shapiro theorem with simple and interesting applications to the Ramanujan tau function and Euler\u27s totient function, the latter of which generalizes an old result of Erdős and Pomerance which shows that as an arithmetic function, the total number of prime factors of values of Euler\u27s totient function satisfies a Gaussian law