Related papers: Minimal gap for higher dimensional sequences
We study the minimal gap statistic for fractional parts of sequences of the form $\mathcal A^\alpha = \{\alpha a(n)\}$ where $\mathcal A = \{a(n)\}$ is a sequence of distinct of integers. Assuming that the additive energy of the sequence is…
We study the minimal gap statistic for sequences of the form $\left( \alpha x_n \right)_{n = 1}^{\infty}$ where $\left( x_n \right)_{n = 1}^{\infty}$ is a sequence of real numbers, and its connection to the additive energy of $\left( x_n…
Let $(a_n)_{n \geq 1}$ be a sequence of distinct positive integers. In a recent paper Rudnick established asymptotic upper bounds for the minimal gaps of $\{a_n \alpha \bmod 1, 1 \leq n \leq N\}$ as $N \to \infty$, valid for Lebesgue-almost…
Let $(a_n)_{n \geq 1}$ be a sequence of distinct positive integers. The metric theory of minimal gaps for the sequence $\{\alpha a_n \text{ mod }1, 1\leq n \leq N\}$ as $N \to \infty$ was initiated by Rudnick, who established that the…
Let \({\mathbb K}\) be any field, let \(X\subset {\mathbb P}^{k-1}\) be a set of \(n\) distinct \({\mathbb K}\)-rational points, and let \(a\geq 1\) be an integer. In this paper we find lower bounds for the minimum distance \(d(X)_a\) of…
Recently, generalizations of the classical Three Gap Theorem to higher dimensions attracted a lot of attention. In particular, upper bounds for the number of nearest neighbor distances have been established for the Euclidean and the maximum…
Following recent work by Koll\'{a}r and Sarnak, we study gaps in the spectra of large connected cubic and quartic graphs with minimum spectral gap. We focus on two sequences of graphs, denoted $\Delta_n$ and $\Gamma_n$ which are more…
Let $\mathbb K$ be a field of characteristic 0. Let $\Gamma\subset\mathbb P^n_{\mathbb K}$ be a reduced finite set of points, not all contained in a hyperplane. Let $hyp(\Gamma)$ be the maximum number of points of $\Gamma$ contained in any…
We study the size of the minimal gap between the first N eigenvalues of the Laplacian on a rectangular billiard having irrational squared aspect ratio $\alpha$, in comparison to the corresponding quantity for a Poissonian sequence. If…
In this paper, we show a new upper bound of prime gaps, that is the gap between a prime number and its consecutive prime number. We show that the gap between a prime number $p_n$ and its consecutive prime number is not larger than…
The three gap theorem, also known as the Steinhaus conjecture or three distance theorem, states that the gaps in the fractional parts of $\alpha,2\alpha,\ldots, N\alpha$ take at most three distinct values. Motivated by a question of…
We discuss upper and lower bounds for the size of gaps in the length spectrum of negatively curved manifolds. For manifolds with algebraic generators for the fundamental group, we establish the existence of exponential lower bounds for the…
This paper deals with the distribution of $\alpha \zeta^{n} \bmod 1$, where $\alpha\neq 0,\zeta>1$ are fixed real numbers and $n$ runs through the positive integers. Denote by $\Vert.\Vert$ the distance to the nearest integer. We…
This paper studies the size of the minimal gap between any two consecutive eigenvalues in the Dirichlet and in the Neumann spectrum of the standard Laplace operator on the Sierpinski gasket. The main result shows the remarkable fact that…
The discrepancy of a binary string is the maximum (absolute) difference between the number of ones and the number of zeroes over all possible substrings of the given binary string. In this note we determine the minimal discrepancy that a…
The simple interpretation of the minimum distance of a linear code obtained by De Boer and Pellikaan, and later refined by the second author, is further developed through the study of various finitely generated graded modules. We use the…
The three gap theorem (or Steinhaus conjecture) asserts that there are at most three distinct gap lengths in the fractional parts of the sequence $\alpha,2\alpha,\ldots,N\alpha$, for any integer $N$ and real number $\alpha$. This statement…
In this paper several examples of gaps (lacunes) between dimensions of maximal and submaximal symmetric models are considered, which include investigation of number of independent linear and quadratic integrals of metrics and counting the…
We provide bounds on the sizes of the gaps -- defined broadly -- in the set $\{k_1\beta_1 + \ldots + k_n\beta_n \mbox{ (mod 1)} : k_i \in \mathbb Z \cap (0,Q^\frac{1}{n}]\}$ for generic $\beta_1, \ldots, \beta_n \in \mathbb R^m$ and all…
Assuming the Riemann hypothesis, this article discusses a new elementary argument that seems to prove that the maximal prime gap of a finite sequence of primes p_1, p_2, ..., p_n <= x, satisfies max {p_(n+1) - p_n : p_n <= x} <=…