Mathematika最新文献

Given a norm ν on $�R^2$, the set of ν-Dirichlet improvable numbers $�{\mathrm{DI}}_{\nu }$ was defined and studied in the papers (Andersen and Duke, Acta Arith. 198 (2021) 37–75 and Kleinbock and Rao, Internat. Math. Res. Notices 2022 (2022) 5617–5657). When ν is the supremum norm, $��{\mathrm{DI}}_{\nu }�=�\mathrm{BA}�\cup �Q�$, where $�\mathrm{BA}$ is the set of badly approximable numbers. Each of the sets $�{\mathrm{DI}}_{\nu }$, like $�\mathrm{BA}$, is of measure zero and satisfies the winning property of Schmidt. Hence for every norm ν, $��\mathrm{BA}�\cap �{\mathrm{DI}}_{\nu }�$ is winning and thus has full Hausdorff dimension. In this article, we prove the following dichotomy phenomenon: either $��\mathrm{BA}�\subset �{\mathrm{DI}}_{\nu }�$ or else $��\mathrm{BA}�\setminus �{\mathrm{DI}}_{\nu }�$

We establish lower bounds for the discrete 2kth moment of the derivative of the Riemann zeta function at nontrivial zeros for all under the Riemann hypothesis and the assumption that all zeros of $��\zeta �\left(�s�\right)�$ are simple.

We study the moments of L-functions associated with primitive cusp forms, in the weight aspect. In particular, we obtain an asymptotic formula for the twisted moments of a long Dirichlet polynomial with modular coefficients. This result, which is conditional on the Generalized Lindelöf Hypothesis, agrees with the prediction of the recipe by Conrey, Farmer, Keating, Rubinstein and Snaith.

If Z is an open subscheme of $��Spec�Z�$, X is a sufficiently nice Z-model of a smooth curve over $�Q$, and p is a closed point of Z, the Chabauty–Kim method leads to the construction of locally analytic functions on $��X�\left(�Z_p�\right)�$ which vanish on $��X�\left(�Z�\right)�$; we call such functions “Kim functions”. At least in broad outline, the method generalizes readily to higher dimensions. In fact, in some sense, the surface M_{0, 5} should be easier than the previously studied curve $��M_{�0�,�4�}�=�P^1�\setminus ��\{�0�,�1�,�\infty �\}��$ since its points are closely related to those of M_{0, 4}, yet they face a further condition to integrality. This is mirrored by a certain weight advantage we encounter, because of which, M_{0, 5} possesses new Kim functions not coming from M_{0, 4}. Here we focus on the case “$��$

We investigate the question whether a scattered compact topological space K such that $��C�\left(�K�\right)�$ has a norming Markushevich basis (M-basis, for short) must be Eberlein. This question originates from the recent solution, due to Hájek, Todorčević and the authors, to an open problem from the 1990s, due to Godefroy. Our prime tool consists in proving that $��C�\left(��[�0�,�{\omega }_1�]��\right)�$ does not embed in a Banach space with a norming M-basis, thereby generalising a result due to Alexandrov and Plichko. Subsequently, we give sufficient conditions on a compact K for $��C�\left(�K�\right)�$ not to embed in a Banach space with a norming M-basis. Examples of such conditions are that K is a zero-dimensional compact space with a P-point, or a compact tree of height at least $��{\omega }_1�+�1�$. In particular, this allows us to answer the said question in the case when K is a tree and to obtain a rather general result for Valdivia compacta. Finally, we give some structural results for scattered compact trees; in particular, we prove that scattered trees of height less than ω₂ are Valdivia.

Given an integer $��b�⩾�2�$ and a set $�P$ of prime numbers, the set $�T_P$ of Toeplitz numbers comprises all elements of [0, b[ whose digits $�{�\left(�a_n�\right)�}_{�n�⩾�1�}$ in the base-b expansion satisfy $��a_n�=�a_{�p�n�}�$ for all $��p�\in �P�$ and $��n�⩾�1�$. Using a completely additive arithmetical function, we construct a number in $�T_P$ that is simply Borel normal if, and only if, $���{\sum }_{�p�\notin �}$

We study the density of solutions to Diophantine inequalities involving non-singular ternary forms, or equivalently, the density of rational points close to non-singular plane algebraic curves.

In this paper, we obtain the functional Orlicz–Brunn–Minkowski inequality and the functional Orlicz–Minkowski inequality for q-torsional rigidity in the smooth category. Furthermore, using an approximation method, we give the general functional Orlicz–Brunn–Minkowski inequality for q-torsional rigidity. As a corollary, we reveal that the functional Orlicz–Brunn–Minkowski inequality is equivalent to the functional Orlicz–Minkowski inequality for q-torsional rigidity in the smooth category. We also give some applications with respect to these two inequalities.

We consider random symmetric matrices with independent entries distributed according to the Haar measure on $�Z_p$ for odd primes p and derive the distribution of their canonical form with respect to several equivalence relations. We give a few examples of applications including an alternative proof for the result of Bhargava, Cremona, Fisher, Jones and Keating on the probability that a random quadratic form over $�Z_p$ has a non-trivial zero.