Mathematika最新文献

In 2004, de Mathan and Teulié stated the $$-adic Littlewood conjecture ($$-LC) in analogy with the classical Littlewood conjecture. Let $$ be a finite field $$ be an irreducible polynomial with coefficients in $$. This paper deals with the analogue of $$-LC over the ring of formal Laurent series over $$, known as the $$-adic Littlewood conjecture ($$-LC).

First, it is shown that any counterexample to $$-LC for the case $$ induces a counterexample to $$-LC when $$ is any irreducible polynomial. Since Adiceam, Nesharim and Lunnon (2021) disproved $$-LC when $$ and when $$ is a finite field with characteristic 3, one obtains a disproof of $$-LC over any such field in full generality (i.e., for any choice of irreducible polynomial $$).

The remainder of the paper is dedicated to proving two metric results on $$-LC with an additional monotonic growth function $$ over an arbitrary finite field. The first — a Khintchine-type theorem for $$-adic multiplicative approximation — enables one to determine the measure of the set of counterexamples to $$-LC for any choice of $$. The second complements this by showing that the Hausdorff dimension of the same set is maximal in the critical case where $$. These results are in agreement with the corresponding theory of multiplicative Diophantine approximation over the reals.

Beyond the originality of the results, the main novelty of the work comes from the methodology used. Classically, Diophantine approximation employs methods from either Number Theory or Ergodic Theory. This paper provides a third option: combinatorics. Specifically, an extensive combinatorial theory is developed relating $$-LC to the properties of the so-called number wall of a sequence. This is an infinite array containing the determinant of every finite Toeplitz matrix generated by that sequence. In full generality, the paper creates a dictionary allowing one to transfer statements in Diophantine approximation in positive characteristic to combinatorics through the concept of a number wall, and conversely.

We obtain the modular automorphism group of any quotient modular curve of level $$, with $$. In particular, we obtain some unexpected automorphisms of order 3 that appear for the quotient modular curves when the Atkin–Lehner involution $$ belongs to the quotient modular group. We also prove that such automorphisms are not necessarily defined over $$. As a consequence of these results, we obtain the full automorphism group of the quotient modular curve $$, for sufficiently large $$.

A classical consequence of the John Ellipsoid Theorem is the upper bound $$ on the Banach–Mazur distance between the Euclidean ball and any symmetric convex body in $$. Equality is attained for the parallelotope and the cross-polytope. While it is known that they are unique with this property for $$ but not for $$, no proof of the characterization of the three-dimensional equality case seems to have ever been published. We fill this gap by showing that the parallelotope and the cross-polytope are the unique maximizers for $$. Our proof is based on an extension of a characterization of distance ellipsoids due to Ader from 1938, which predates the John Ellipsoid Theorem. Ader's characterization turns out to provide a decomposition similar to the John decomposition, which leads to a proof of the aforementioned $$ estimate that bypasses the concept of volumes and reveals precise information about the equality case. We highlight further consequences of Ader's characterization, including a proof of an unpublished result attributed to Maurey related to the uniqueness of distance ellipsoids. In addition, we investigate more closely the role of the parallelogram as a maximizer in various problems related to the distance between planar symmetric convex bodies. We establish the stability of the parallelogram as the unique planar symmetric convex body with the maximal distance to the Euclidean disk with the best possible (linear) order. This uniqueness extends to the setting of pairs of planar 1-symmetric convex bodies, where we show that the maximal possible distance between them is again $$, together with a characterization of the equality case involving the parallelogram.

Let $$ be a large prime and let $$. We prove that if $$ is a $$-valued multiplicative function, such that the exponential sums

In the “Covering” pursuit game on a graph, a robber and a set of cops play alternately, with the cops each moving to an adjacent vertex (or not moving) and the robber moving to a vertex at distance at most 2 from his current vertex. The aim of the cops is to ensure that, after every one of their turns, there is a cop at the same vertex as the robber. How few cops are needed? Our main aim in this paper is to consider this problem for the two-dimensional grid $$. Bollobás and Leader asked if the number of cops needed is $$. We answer this question by showing that $$ cops suffice. We also consider some applications. In particular we study the game “Catching a Fast Robber,” concerning the number of cops needed to catch a fast robber of speed $$ on the two-dimensional grid $$. We improve the bounds proved by Balister, Bollobás, Narayanan and Shaw for this game.

We prove that every $$-vertex linear triple system with $$ edges has at least $$ copies of a pentagon, provided $$. This provides the first nontrivial bound for a question posed by Jiang and Yepremyan. More generally, for each $$, we prove that there is a constant $$ such that if an $$-vertex graph is $$-far from being triangle-free, with $$, then it has at least $$ copies of $$. This improves the previous best bound of $$ due to Gishboliner, Shapira, and Wigderson. Our result also yields some geometric theorems, including the following. For $$ large, every $$-point set in the plane with at least $$ triangles similar to a given triangle $$, contains two triangles sharing a special point, called the harmonic point. In the other direction, we give a construction showing that the exponent $$ cannot be reduced to anything smaller than $$ and improve this further to $$ for a 3-partite version of the problem.

Szüsz's inhomogeneous version (1958) of Khintchine's theorem (1924) gives conditions on $$ under which for almost every real number $$ there exist infinitely many rationals $$ such that

The celebrated Artin conjecture on primitive roots asserts that given any integer $$ that is neither $$ nor a perfect square, there is an explicit constant $$ such that the number $$ of primes $$ for which $$ is a primitive root is asymptotically $$ as $$, where $$ counts the number of primes not exceeding $$. Artin's conjecture has remained unsolved since its formulation in 1927. Nevertheless, Hooley demonstrated in 1967 that Artin's conjecture is a consequence of the Generalized Riemann Hypothesis (GRH) for Dedekind zeta functions of certain cyclotomic-Kummer extensions over $$. In this paper, we use GRH to establish a uniform version of the Artin–Hooley asymptotic formula. Specifically, we prove that $$ whenever $$, that is, whenever $$ tends to infinity faster than any power of $$. Under GRH, we also show that the least prime $$ possessing $$ as a primitive root satisfies the upper bound $$ uniformly for all nonsquare $$. We conclude with an application to the average value of $$ and a discussion of an analog concerning the least “almost-primitive” root.

We consider two well-known problems: upper bounding the volume of lower dimensional ellipsoids contained in convex bodies given their John ellipsoid, and lower bounding the volume of ellipsoids containing projections of convex bodies given their Loewner ellipsoid. For the first problem, we use the John asymmetry to unify a tight upper bound for the general case by Ball with a stronger inequality for symmetric convex bodies. We obtain an inequality that is tight for most asymmetry values in large dimensions and an even stronger inequality in the planar case that is always best possible. In contrast, we show for the second problem an inequality that is tight for bodies of any asymmetry, including cross-polytopes, parallelotopes, and (in almost all cases) simplices. Finally, we derive some consequences for the width-circumradius- and diameter-inradius-ratios when optimized over affine transformations and show connections to the Banach–Mazur distance.