Tag - Number theory
A central question in additive combinatorics is to understand how large arithmetic progression-free sets can be. In this talk, I will focus on this question for high-dimensional generalization of arithmetic progressions (AP) known as corners. A (2-dimensional) corner is a triple of the form (x,y),(x+d,y),(x,y+d) for some d>0 in [N] × [N]. Extending this definition to higher dimensions, a k-dimensional corner in [N]k is a (k+1)-tuple defined similarly for some d. While it is known that corner-free sets have a vanishingly small density, the precise bounds on their size remain unknown. Until recently, the best-known corner-free sets were derived from constructions of AP-free sets: a construction of a 3-term AP-free set by Behrend from 1946, and a generalization by Rankin for k-term APs in 1961. New results by Linial and Shraibman (CCC 2021) and Green (New Zealand Journal of Mathematics 2021) changed this picture; they improved the upper bound for k=2 by adopting a communication complexity point of view.
I will discuss our recent work where we employ the same perspective of communication complexity and obtain the first improvement on the upper bound of the size of high-dimensional (k>2) corner-free sets since the original construction of Rankin.
In this talk, I will present some results on the class field theory of smooth projective curves over a local field where one allows arbitrary ramification along a proper closed subset. We shall derive these results using some new results on the class field theory of 2-local fields and a duality theorem. This is based on a joint work with Subhadip Majumder.
We give birational parametrizations of pairs (C,T) where T is a 5-torsion point on the Jacobian of a genus-2 curve C, possibly satisfying one or more of the following additional conditions:
• T = (P) - (P0) for some points P0, P in C with P0 Weierstrass;
• C has one or more additional rational Weierstrass points;
• Jac(C) has real multiplication by (1+√5)/2, and T is a √5-torsion point.
L-functions of degree d can be parametrized, in two different ways, by points with an attached multiplicity in (d-1)-dimensional Euclidean space. One approach separates the L-functions according to the shape of the Gamma-factors in the functional equation, equivalently, according to the infinity type of the underlying automorphic representation. The other approach combines all the L-functions of a given degree into a single picture in which the points, to leading order, are uniformly dense. We will describe these classifications and provide examples of several 'landscapes' in the L-function world.
We are going to survey some recent and less recent results pertaining to the study of finite point configurations in Euclidean space and vector spaces over finite fields, centred around the Erdős/Falconer distance problems. We shall place particular emphasis on number-theoretic ideas and obstructions that arise in this area.
In a joint work with Alexei Skorobogatov we used a second-moment approach to prove asymptotics for the average of the von Mangoldt function over the values of a typical integer polynomial. As a consequence, we proved Schinzel's Hypothesis in 100% of the cases. In addition, we proved that a positive proportion of Châtelet equations have a rational point. I will explain subsequent joint work with Tim Browning and Joni Teräväinen that develops the method and establishes asymptotics for averages of an arithmetic function over the values of typical polynomials. Part of the new ideas come from the theory of averages of arithmetic functions in short intervals. One of the applications is that the Hasse principle holds for 100% of Châtelet equations. This agrees with the conjecture of Colliot-Thélène stating that the Brauer-Manin obstruction is the only obstruction to the Hasse principle for rationally connected varieties.
In this talk, we shall explore certain polygonal paths, that we call ''Legendre paths'', which encode important information about the values of the Legendre symbol. More precisely, the Legendre path modulo a prime number p is defined as the polygonal path in the plane whose vertices are the points (j, Sp(j)) for 0≤j≤p-1, where Sp(j) is the (normalized) sum of Legendre symbols (n/p) for n up to j. In particular, we will attempt to answer the following questions as we vary over the primes p: how are these paths distributed? how do their maximums behave? when does a Legendre path decreases for the first time? what is the typical number of x-intercepts of such paths? and what proportion of a Legendre path is above the real axis? We will see that some of these questions correspond to important and longstanding problems in analytic number theory, including understanding the size of the least quadratic non-residue, and improving the Pólya-Vinogradov inequality for character sums. Among our results, we prove that as we average over the primes, the Legendre paths converge in law, in the space of continuous functions, to a certain random Fourier series constructed using Rademacher random multiplicative functions. Part of this work is joint with Ayesha Hussain and with Oleksiy Klurman and Marc Munsch.
A conjecture of Chowla postulates that no L-function of Dirichlet characters over the rationals vanishes at s=1/2. Soundararajan has proved non-vanishing for a positive proportion of quadratic characters. Over function fields Li has discovered that Chowla's conjecture fails for infinitely many distinct quadratic characters. However, on the basis of the Katz-Sarnak heuristics, it is still widely believed that one should have non-vanishing for 100% of the characters in natural families (such as the family of quadratic characters). Works of Bui-Florea, David-Florea-Lalin, Ellenberg-Li-Shusterman, among others, provided evidence giving a positive proportion of non-vanishing in several such families. I will present an upcoming joint work with Peter Koymans and Mark Shusterman, where we prove that for each fixed q congruent to 3 modulo 4 one has 100% non-vanishing in the family of imaginary quadratic function fields.
Given a set ℬ of natural numbers, not containing 1, we denote by ℱℬ the set of ℬ-free numbers, that is, ℱℬ = ℤ \ ⋃b∈ℬ bℤ. Let Xη be the ℬ-free subshifts, that is the subshift induced by η, where η denotes the characteristic function of ℱℬ . That means, Xη is the closure of the set of all shifts of η in the space {0, 1}ℤ equipped with the product topology. We are interested in the case when B is a Behrend set, that is, when the set of ℬ-free numbers has zero density. It turns out that this is the case precisely when Xη is proximal and has zero entropy. We prove that the complexity of Xη, with ℬ being a Behrend set, can achieve any subexponential growth. Together with the ℬ-free shift we investigate the ℬ-admissible shift Xℬ and we show that it is transitive if and only if the set B is pairwise coprime, which allows one to characterize dynamically the subshifts generated by the Erdős sets (infinite, coprime and not Behrend). We also estimate the complexity for some classical subshifts (the subshift of primes or semi-primes). The lower estimates are obtained conditionally on Hardy-Littlewood Conjecture or Dickson’s Conjecture. We remark on a recent result of Tao and Ziegler (not assuming the conjectures) that the shift of primes is uncountable.
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