Mathematika最新文献

For every irrational real $$, let $$ denote the largest partial quotient in its continued fraction expansion (or $$, if unbounded). The 2-adic Littlewood conjecture (2LC) can be stated as follows: There exists no irrational $$ such that $$ is uniformly bounded by a constant $$ for all $$. In 2016, Badziahin proved (considering a different formulation of 2LC) that if a counterexample exists, then the bound $$ is at least 8. We improve this bound to 15. Then we focus on a “B-variant” of 2LC, where we replace $$ by $$. In this setting, we prove that if $$ for all $$, then $$. For the proof we use Hurwitz's algorithm for multiplication of continued fractions by 2. Along the way, we find families of quadratic irrationals $$ with the property that for arbitrarily large $$ there exist $$ all equivalent to $$.

In 2021, Ordentlich, Regev, and Weiss made a breakthrough that the lattice covering density of any $$-dimensional convex body is upper bounded by $$, improving on the best previous bound established by Rogers in 1959. However, for the Euclidean ball, Rogers obtained the better upper bound $$, and this result was extended to certain symmetric convex bodies by Gritzmann. The constant $$ above is independent on $$. In this paper, we show that such a bound can be achieved for more general classes of convex bodies without symmetry, including anti-blocking bodies, locally anti-blocking bodies and $$-dimensional polytopes with $$ vertices.

We prove a conjecture of R. Oberlin and Héra on the dimension of unions of $$-planes. Let $$ be integers, and $$. If $$, with $$, then $$. The proof combines a recent idea of Zahl and the Brascamp–Lieb inequality.

Pál's isominwidth theorem states that for a fixed minimal width, the regular triangle has minimal area. A spherical version of this theorem was proven by Bezdek and Blekherman, if the minimal width is at most $$. If the width is greater than $$, the regular triangle no longer minimizes the area at fixed minimal width. We show that the minimizers are instead given by the polar sets of spherical Reuleaux triangles. Moreover, stability versions of the two spherical inequalities are obtained.

The Brjuno and Wilton functions bear a striking resemblance, despite their very different origins; while the Brjuno function $$ is a fundamental tool in one-dimensional holomorphic dynamics, the Wilton function $$ stems from the study of divisor sums and self-correlation functions in analytic number theory. We show that these perspectives are unified by the semi-Brjuno function $$. Namely, $$ and $$ can be expressed in terms of the even and odd parts of $$, respectively, up to a bounded defect. Based on numerical observations, we further analyze the arising functions $$ and $$, the first of which is Hölder continuous whereas the second exhibits discontinuities at rationals, behaving similarly to the classical popcorn function.

For all $$ with $$, the smallest possible isoperimetric quotient of an $$-dimensional convex polytope that has $$ facets is shown to be bounded from above and from below by positive universal constant multiples of $$. For all $$ and $$, it is shown that every $$-dimensional origin-symmetric convex polytope that has $$ vertices admits an affine image whose isoperimetric quotient is at most a universal constant multiple of $$, which is sharp. The weak isomorphic reverse isoperimetry conjecture is proved for $$-dimensional convex polytopes that have $$ facets by demonstrating that any such polytope $$ has an image $$ under a volume-preserving matrix and a convex body $$ such that the isoperimetric quotient of $$ is at most a universal constant multiple of $$, and also $$ is at least a positive universal constant.

We show that there exists a lattice covering of $$ by Euclidean spheres of equal radius with density $$ as $$, where

We study the low-lying zeros of a family of $$-functions attached to the complex multiplication elliptic curve $$, for each odd and square-free integer $$. Specifically, upon writing the $$-function of $$ as $$ for the appropriate Grössencharakter $$ of conductor $$, we consider the collection $$ of $$-functions attached to $$, $$, where for each integer $$, $$ denotes the primitive character inducing $$. We observe that $$ of the $$-functions in $$ have negative root number. $$ is thus not one of the essentially homogeneous families of the universality conjecture of Sarnak, Shin and Templier [33], with unitary, symplectic or orthogonal (odd or even) symmetry type. By computing the one-level density in the family of $$-functions in $$ with conductor at most $$, we find that $$ naturally decomposes into subfamilies: more specifically, a collection of symplectic ($$ for $$, $$ even) and orthogonal ($$ for $$, $$ odd) subfamilies. For each such subfamily, we moreover compute explicit lower order terms in decreasing powers of $$.

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 $$, for sufficiently large $$.