Mathematika最新文献

Let $$ and $$ be the algebra of all bounded linear operators on a complex Hilbert space $$ and the Jordan algebra of all self-adjoint operators in $$, respectively. In this paper, we give characterizations of rank one operators by the pseudospectrum on $$-Lie product of bounded linear operators and discuss some properties about the pseudospectrum. As applications, we obtain the structures of all surjective maps preserving the pseudospectrum of $$-Lie product on $$ and $$, respectively.

The combinatorial discrepancy of arithmetic progressions inside $$ is the smallest integer $$ for which $$ can be colored with two colors so that any arithmetic progression in $$ contains at most $$ more elements from one color class than the other. Bounding the discrepancy of such set systems is a classical problem in the discrepancy theory. More recently, this problem was generalized to arithmetic progressions in grids like $$ (Valkó) and $$ (Fox, Xu, and Zhou). In the latter setting, Fox, Xu, and Zhou gave upper and lower bounds on the discrepancy that match within a $$ factor, where $$ is the ground set. In this work, we use the connection between factorization norms and discrepancy to improve their upper bound to be within a $$ factor from the lower bound. We also generalize Fox, Xu, and Zhou's lower bound, and our upper bounds to arithmetic progressions in arbitrary convex bodies.

A classical theorem of Macbeath states that for any integers $$, $$, $$-dimensional Euclidean balls are hardest to approximate, in terms of volume difference, by inscribed convex polytopes with $$ vertices. In this paper, we investigate normed variants of this problem: we intend to find the extremal values of the Busemann volume, Holmes–Thompson volume, Gromov's mass, and Gromov's $$ of a largest volume convex polytope with $$ vertices, inscribed in the unit ball of a $$-dimensional normed space.

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