Mathematika最新文献

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

Let $$ be a subharmonic function on the open unit disc $$, centered at the origin of the complex plane, and let $$ be a holomorphic function such that $$. A classical result, known as Littlewood subordination principle, states $$, where $$ and $$ are integral means over the circle of radius $$ centered at the origin, of the functions $$ and $$, respectively. In this note, we obtain an unexpected improvement of Littlewood subordination principle in the case when the function $$ is univalent, by proving that

Let $$. We prove an unconditional lower bound on the measure of the sets $$ for $$. For $$, our bound has a Gaussian shape with variance proportional to $$. At the endpoint, $$, our result implies the best known $$-theorem for $$ that is due to Tsang. We also explain how the method breaks down for $$ given our current knowledge about the zeros of the zeta function. Conditionally on the Riemann hypothesis, we extend our results to the range $$.

For the Lagrange spectrum and other applications, we determine the smallest accumulation point of binary sequences that are maximal in their shift orbits. This problem is trivial for the lexicographic order, and its solution is the fixed point of a substitution for the alternating lexicographic order. For orders defined by cylinders, we show that the solutions are $$-adic sequences, where $$ is a certain infinite set of substitutions that contains Sturmian morphisms. We also consider a similar problem for symmetric ternary shifts, which is applicable to the multiplicative version of the Markoff–Lagrange spectrum.

We prove multiple vector-valued and mixed-norm estimates for multilinear operators in $$, more precisely for multilinear operators $$ associated to a symbol singular along a $$-dimensional space and for multilinear variants of the Hardy-Littlewood maximal function. When the dimension $$, the input functions are not necessarily in $$ and can instead be elements of mixed-norm spaces $$.

Such a result has interesting consequences especially when $$ spaces are involved. Among these, we mention mixed-norm Loomis-Whitney-type inequalities for singular integrals, as well as the boundedness of multilinear operators associated to certain rational symbols. We also present examples of operators that are not susceptible to isotropic rescaling, which only satisfy “purely mixed-norm estimates” and no classical $$ estimates.

Relying on previous estimates implied by the helicoidal method, we also prove (non-mixed-norm) estimates for generic singular Brascamp-Lieb-type inequalities.