Mathematika最新文献

In this paper, we establish different sharp forms of Mahler's conjecture for $$-concave even functions in dimensions $$, for $$ and 2, for $$, thus generalizing our previous results in Fradelizi and Nakhle (Int. Math. Res. Not. 12 (2023), 10067–10097) on log-concave even functions in dimension 2, which corresponds to the case $$. The functional volume product of an even $$-concave function $$ is

Let $$ be the $$ matrix algebra over $$ and $$ be the invertible elements in $$. Inspired by Kaplansky–Lv́ov conjecture, we explore the image of polynomials with constants, namely polynomials from the free algebra $$. In this article, we compute the images of the polynomial maps given by (a) generalized sum of powers $$ and (b) generalized commutator map $$, where $$, $$ are nonzero elements of $$ when $$ is an algebraically closed field. We show that the images of these maps are vector spaces. For the polynomial in (a), we compute the images by fixing a simultaneous conjugate pair for $$, and it turns out that it is surjective in most cases.

The $$ Theorem for $$ asserts that, if $$ are finite, nonempty subsets with $$ and $$, then there exist arithmetic progressions $$ and $$ of common difference such that $$ and $$ for all $$. These are instances of Freiman's theorem with precise bounds. There is much partial progress extending this result to nonempty subsets $$ with $$ prime, $$ and $$. The ideal conjectured density restriction under which such a version of the $$ Theorem modulo $$ is expected is $$. Under this ideal density constraint, we show that there are arithmetic progressions $$, $$, and $$ of common difference with $$ and $$ for all $$, where $$, provided $$. This generalizes a result of Serra and Zémor [33] by extending their work from the special case $$ to that of general sumsets $$, removes all unnecessary sufficiently large $$ restrictions, and improves (even in the case $$) their constant 100-fold, from 0.0001 to 0.01. As part of the proof, we additionally obtain a yet better 1000-fold improvement of their constants at the cost of a near optimal density restriction of the form $$ (Theorem 3.5 and Corollary 3.7). These give rare high-density versions of the $$ Theorem for general sumsets $$ modulo $$ and are the first instances with tangible (rather than effectively existential) values for the constants for general sumsets $$ with high density, or indeed for any density without added constraints on the relative sizes of $$ and $$.

Kuperberg and Lalín stated some conjectures on the variance of certain sums of the divisor function $$ over number fields, which were inspired by analogous results over function fields proven by the authors. These problems are related to certain symplectic matrix integrals. While the function field results can be directly related to the random matrix integrals, the connection between the random matrix integrals and the number field results is less direct and involves arithmetic factors. The goal of this article is to give heuristic arguments for the formulas of these arithmetic factors.

Recently, we showed that global root numbers of modular forms are biased toward $$. Together with Pharis, we also showed an initial bias of Fourier coefficients toward the sign of the root number. First, we prove analogous results with respect to local root numbers. Second, a subtle correlation between Fourier coefficients and global root numbers, termed murmurations, was recently discovered for elliptic curves and modular forms. We conjecture murmurations in a more general context of different (possibly empty) combinations of local root numbers. Last, the Appendix corrects a sign error in our joint paper with Pharis.

We construct maximal $$-subsets on a large class of curved manifolds, in an optimal range of Lebesgue exponents $$. Our arguments combine restriction estimates and decoupling with old and new probabilistic estimates.

Let $$ denote the integer part of $$. In 1947, Mills constructed a real number $$ such that $$ is always a prime number for every positive integer $$. We define Mills' constant as the smallest real number $$ satisfying this property. Determining whether this number is irrational has been a long-standing problem. In this paper, we show that Mills' constant is irrational. Furthermore, we obtain partial results on the transcendency of this number.

We consider a sequence $$ of polynomials with uniformly bounded zeros and $$, $$ for $$, satisfying certain asymptotic conditions. We prove that the function sequence $$ is uniformly convergent in $$. The non-autonomous filled Julia set $$ generated by the polynomial sequence $$ is defined and shown to be compact and regular with respect to the Green function. Our toy example is generated by $$, where $$ is the classical Chebyshev polynomial of degree $$.

In this paper, we introduce o-extreme points defined by using orthogonal paths in orthogonally connected sets. We investigate their properties and obtain Minkowski-type theorems involving orthogonally connected sets. Using o-extreme points, we give some characterizations of staircase connectedness.

We consider the set of points in infinitely many max-norm annuli centred at rational points in $$. We give Jarník–Besicovitch-type theorems for this set in terms of Hausdorff dimension. Interestingly, we find that if the outer radii are decreasing sufficiently slowly, dependent only on the dimension $$, and the thickness of the annuli is decreasing rapidly, then the dimension of the set tends towards $$. We also consider various other forms of annuli including rectangular annuli and quasi-annuli described by the difference between balls of two different norms. Our results are deduced through a novel combination of a version of Cassel's scaling lemma and a generalisation of the Mass Transference Principle, namely the Mass transference principle from rectangles to rectangles due to Wang and Wu (Math. Ann. 2021).