Mathematika最新文献

Let $$ be a convex body in $$ in which a ball rolls freely and which slides freely in a ball. Let $$ be the intersection of $$ i.i.d. random half-spaces containing $$ chosen according to a certain prescribed probability distribution. We prove an asymptotic upper bound on the variance of the mean width of $$ as $$. We achieve this result by first proving an asymptotic upper bound on the variance of the weighted volume of random polytopes generated by $$ i.i.d. random points selected according to certain probability distributions, then, using polarity, we transfer this to the circumscribed model.

We give a construction of a convex set $$ with cardinality $$ such that $$ contains a convex subset with cardinality $$. We also consider the following variant of this problem: given a convex set $$, what is the size of the largest matching $$ such that the set

Let $$ be a matrix. In this paper, we investigate the set $$ of badly approximable targets for $$, where $$ is the $$-torus. It is well known that $$ is a winning set for Schmidt's game and hence is a dense subset of full Hausdorff dimension. We investigate the relationship between the measure of $$ and Diophantine properties of $$. On the one hand, we give the first examples of a nonsingular $$ such that $$ has full measure with respect to some nontrivial algebraic measure on the torus. For this, we use transference theorems due to Jarnik and Khintchine, and the parametric geometry of numbers in the sense of Roy. On the other hand, we give a novel Diophantine condition on $$ that slightly strengthens nonsingularity, and show that under the assumption that $$ satisfies this condition, $$ is a null-set with respect to any nontrivial algebraic measure on the torus. For this, we use naive homogeneous dynamics, harmonic analysis, and a novel concept that we refer to as mixing convergence of measures.

Let

Let $$ be a prime power and $$ be the rational function field over $$, the field with $$ elements. Let $$ be a Drinfeld module over $$ and $$ be a nonzero prime ideal of $$. Over the constant $$-extension of $$, we introduce the fine Selmer group associated to the $$-primary torsion of $$. We show that it is a cofinitely generated module over $$. This proves an analogue of Iwasawa's $$ conjecture in this setting, and provides context for the further study of the objects that have been introduced in this article.

A Hajnal–Máté graph is an uncountably chromatic graph on $$ satisfying a certain natural sparseness condition. We investigate Hajnal–Máté graphs and generalizations thereof, focusing on the existence of Hajnal–Máté graphs in models resulting from adding a single Cohen real. In particular, answering a question of Dániel Soukup, we show that such models necessarily contain triangle-free Hajnal–Máté graphs. In the process, we isolate a weakening of club guessing called disjoint-type guessing that we feel is of interest in its own right. We show that disjoint-type guessing is independent of $$ and, if disjoint-type guessing holds in the ground model, then the forcing extension by a single Cohen real contains Hajnal–Máté graphs $$ such that the chromatic numbers of finite subgraphs of $$ grow arbitrarily slowly.

In this paper, we extend the work of Pollington and Velani [Selecta Math. 11(2005)] to an $$-arithmetic set-up, where $$ is a finite set of valuations of $$. In particular, for an absolutely friendly measure $$ supported on a compact set in $$, we give a summation condition on an approximating function $$ such that $$ almost no point in the compact set is $$ approximable. The crucial ingredient is a version of the simplex lemma that we prove dynamically.

We prove that product-free subsets of the free group over a finite alphabet have maximum upper density $$ with respect to the natural measure that assigns total weight one to each set of irreducible words of a given length. This confirms a conjecture of Leader, Letzter, Narayanan, and Walters. In more general terms, we actually prove that strongly $$-product-free sets have maximum upper density $$ in terms of this measure. The bounds are tight.

We introduce the centred and the uncentred triangular maximal operators $$ and $$, respectively, on any locally finite tree in which each vertex has at least three neighbours. We prove that both $$ and $$ are bounded on $$ for every $$ in $$, that $$ is also bounded on $$, and that $$ is not of weak type (1, 1) on homogeneous trees. Our proof of the $$ boundedness of $$ hinges on the geometric approach of Córdoba and Fefferman. We also establish $$ bounds for some related maximal operators. Our results are in sharp contrast with the fact that the centred and the uncentred Hardy–Littlewood maximal operators (on balls) may be unbounded on $$ for every $$ even on some trees where the number of neighbours is uniformly bounded.

Let $$ be a $$-tuple of positive real numbers such that $$ and $$. A $$-dimensional vector $$ is said to be $$-singular if for every $$, there exists $$ such that for all $$, the system of inequalities