Mathematika最新文献

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.

Let $$ be pairwise distinct primes. From a theorem of Kummer, each prime $$ can divide $$ at most $$ times. We show that, for all $$, if $$ are sufficiently large in terms of $$ and $$, then there exist infinitely many positive integers $$ such that each $$ divides $$ at most $$ times. We connect this result to a famous conjecture by Graham on whether there are infinitely many integers $$ such that $$ is coprime to 105.

Let $$ be the Fourier coefficients of an $$ Hecke–Maass cusp form $$ and $$ be those of an $$ Hecke holomorphic or Hecke–Maass cusp form $$. Let $$ and $$ be a sequence. We show that if $$ for some $$,

Determinantal point processes exhibit an inherent repulsive behavior, thus providing examples of very evenly distributed point sets on manifolds. In this paper, we study the so-called harmonic ensemble, defined in terms of Laplace eigenfunctions on the sphere $$ and the flat torus $$, and the so-called spherical ensemble on $$, which originates in random matrix theory. We extend results of Beltrán, Marzo, and Ortega-Cerdà on the Riesz $$-energy of the harmonic ensemble to the nonsingular regime $$, and as a corollary find the expected value of the spherical cap $$ discrepancy via the Stolarsky invariance principle. We find the expected value of the $$ discrepancy with respect to axis-parallel boxes and Euclidean balls of the harmonic ensemble on $$. We also show that the spherical ensemble and the harmonic ensemble on $$ and $$ with $$ points attain the optimal rate $$ in expectation in the Wasserstein metric $$, in contrast to independent and identically distributed random points, which are known to lose a factor of $$.

We establish several convexity properties for the entropy and Fisher information of mixtures of centred Gaussian distributions. Firstly, we prove that if $$ are independent scalar Gaussian mixtures, then the entropy of $$ is concave in $$, thus confirming a conjecture of Ball, Nayar and Tkocz (2016) for this class of random variables. In fact, we prove a generalisation of this assertion which also strengthens a result of Eskenazis, Nayar and Tkocz (2018). For the Fisher information, we extend a convexity result of Bobkov (2022) by showing that the Fisher information matrix is operator convex as a matrix-valued function acting on densities of mixtures in $$. As an application, we establish rates for the convergence of the Fisher information matrix of the sum of weighted i.i.d. Gaussian mixtures in the operator norm along the central limit theorem under mild moment assumptions.

We refine a recent heuristic developed by Keating and the second author. Our improvement leads to a new integral expression for the conjectured asymptotic formula for shifted moments of the Riemann zeta-function. This expression is analogous to a formula, recently discovered by Brad Rodgers and Kannan Soundararajan, for moments of characteristic polynomials of random matrices from the unitary group.

We study the zeros of cusp forms of large weight for the modular group, which have a very large order of vanishing at infinity, so that they have a fixed number $$ of finite zeros in the fundamental domain. We show that for large weight the zeros of these forms cluster near $$ vertical lines, with the zeros of a weight $$ form lying at height approximately $$. This is in contrast to previously known cases, such as Eisenstein series, where the zeros lie on the circular part of the boundary of the fundamental domain, or the case of cuspidal Hecke eigenforms where the zeros are uniformly distributed in the fundamental domain. Our method uses the Faber polynomials. We show that for our class of cusp forms, the associated Faber polynomials, suitably renormalized, converge to the truncated exponential polynomial of degree $$.

Assuming the Generalised Riemann Hypothesis, we prove a sharp upper bound on moments of shifted Dirichlet L-functions. We use this to obtain conditional upper bounds on high moments of theta functions. Both of these results strengthen theorems of Munsch, who proved almost sharp upper bounds for these quantities. The main new ingredient of our proof comes from a paper of Harper, who showed the related result $$ for all $$ under the Riemann Hypothesis. Finally, we obtain a sharp conditional upper bound on high moments of character sums of arbitrary length.