Related papers: Gaps between logs

We study the semiclassical quantization of Poincar\'e maps arising in scattering problems with fractal hyperbolic trapped sets. The main application is the proof of a fractal Weyl upper bound for the number of resonances/scattering poles in…

We consider the Grover walk on a finite graph composed of two arbitrary simple graphs connected by one edge, referred to as a bridge. The parameter $\epsilon>0$ assigned at the bridge represents the strength of connectivity: if…

We study the asymptotic distribution, as the volume parameter goes to 1, of the peak (largest part) of finite- or slowly-growing-width cylindric plane partitions weighted by their trace, seam, and volume. There are two natural asymptotic…

For any positive integer $k$, we show that infinitely often, perfect $k$-th powers appear inside very long gaps between consecutive prime numbers, that is, gaps of size $$ c_k \frac{\log p \log_2 p \log_4 p}{(\log_3 p)^2}, $$ where $p$ is…

Let $g$ be a random matrix distributed according to uniform probability measure on the finite general linear group $\mathrm{GL}_n(\mathbb{F}_q)$. We show that $\mathrm{Tr}(g^k)$ equidistributes on $\mathbb{F}_q$ as $n \to \infty$ as long as…

We give an exponential upper bound on the probabilitywith which the denominator of the $n$th convergent in the regular continued fraction expansion stays away from the mean $\frac{n\pi^2}{12\log2}$. The exponential rate is best possible,…

Consider the representation of a rational number as a continued fraction, associated with "odd" Euclidean algorithm. In this paper we prove certain properties for the limit distribution function for sequences of rationals with bounded sum…

This work establishes computable bounds between f-divergences for probability measures within a generalized quasi-$\varepsilon_{(M,m)}$-neighborhood framework. We make the following key contributions. (1) a unified characterization of local…

In the 1990's exponential-type error bounds appeared in the theory of radial basis functions. This kind of error bounds is very powerful. However it only measures the difference between the approximant and approximand. Mathematicians and…

Let $F$ be a univariate polynomial or rational fraction of degree $d$ defined over a number field. We give bounds from above on the absolute logarithmic Weil height of $F$ in terms of the heights of its values at small integers: we review…

We find Weyl upper bounds for the quantum open baker's map in the semiclassical limit. For the number of eigenvalues in an annulus, we derive the asymptotic upper bound $\mathcal O(N^\delta)$ where $\delta$ is the dimension of the trapped…

We consider the GUE minor process, where a sequence of GUE matrices is drawn from the corner of a doubly infinite array of i.i.d. standard normal variables subject to the symmetry constraint. From each matrix, we take its largest…

Given a fixed quadratic extension K of Q, we consider the distribution of elements in K of norm 1 (denoted N). When K is an imaginary quadratic extension, N is naturally embedded in the unit circle in C and we show that it is…

The distributional properties of the translation flow on the unit square have been considered in different fields of mathematics, including algebraic geometry and discrepancy theory. One method to quantify equidistribution is to compare the…

We consider Wald's sequential probability ratio test for deciding whether a sequence of independent and identically distributed observations comes from a specified phase-type distribution or from an exponentially tilted alternative…

We establish a central limit theorem for counting large continued fraction digits $(a_n)$, i.e. we count occurrences $\{a_n>b_n\}$, where $(b_n)$ is a sequence of positive integers. Our result improves a similar result by Philipp which…

We define the crossing number for an embedding of a graph G into R^3, and prove a lower bound on it which almost implies the classical crossing lemma. We also give sharp bounds on the space crossing numbers of pseudo-random graphs.

In this paper we prove a quantiative local limit theorem for the distribution of the number of triangles in the Erd\H{o}s-Renyi random graph $G(n,p)$, for a fixed $p\in (0,1)$. This proof is an extension of the previous work of Gilmer and…

Let $N_n(a, b)$ denote the number of real zeros of Gaussian elliptic polynomials of degree $n$ on the interval $(a, b)$, where $a$ and $b$ may vary with $n$. We obtain a precise formula for the variance of $N_n(a, b)$ and utilize this…

Let $r \ge 2$ be an integer. We adapt the Maynard-Tao sieve to produce the asymptotically best-known bounded gaps between products of $r$ distinct primes. Our result applies to positive-density subsets of the primes that satisfy certain…