Related papers: On the smallest gap in a sequence with Poisson pai…

A deterministic sequence of real numbers in the unit interval is called \emph{equidistributed} if its empirical distribution converges to the uniform distribution. Furthermore, the limit distribution of the pair correlation statistics of a…

We show that at least 1/3 of positive real numbers are in the set of limit points of normalized prime gaps. More precisely, if $p_n$ denotes the $n$th prime and $\mathbb{L}$ is the set of limit points of the sequence $\{(p_{n+1}-p_n)/\log…

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} <=…

Let $p_n$ denote the $n$-th prime. For any $m\geq 1$, there exist infinitely many $n$ such that $p_{n}-p_{n-m}\leq C_m$ for some large constant $C_m>0$, and $$p_{n+1}-p_n\geq \frac{c_m\log n\log\log n\log\log\log\log n}{\log\log\log n}, $$…

In a recent work Friedlander studied the problem of how large consecutive prime gaps should be in order that the sum of the reciprocals should be divergent. Supposing a very deep Hypothesis, a generalization of the Hardy--Littlewood prime…

We consider the two-dimensional Coulomb gas with a general potential at the determinantal temperature, or equivalently, the eigenvalues of random normal matrices. We prove that the smallest gaps between particles are typically of order…

Fix $\alpha,\theta >0$, and consider the sequence $(\alpha n^{\theta} \mod 1)_{n\ge 1}$. Since the seminal work of Rudnick--Sarnak (1998), and due to the Berry--Tabor conjecture in quantum chaos, the fine-scale properties of these dilated…

Suppose that $1<c<9/8$. For any $m\geq 1$, there exist infinitely many $n$ such that $$ \{[n^c],\ [(n+1)^c],\ \ldots,\ [(n+k_0)^c]\} $$ contains at least $m+1$ primes, if $k_0$ is sufficiently large (only depending on $m$).

We showed that the prime gap for a prime number p is less than or equal to the prime count of the prime number.

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…

We study the gaps between consecutive prime numbers directly through Eratosthenes sieve. Using elementary methods, we identify a recursive relation for these gaps and for specific sequences of consecutive gaps, known as constellations.…

Let $\mathcal S=\{s_1<s_2<s_3<\ldots\}$ be the sequence of all natural numbers which can be represented as a sum of two squares of integers. For $X\ge2$ we denote by $g(X)$ the largest gap between consecutive elements of $\mathcal S$ that…

We study the notion of inhomogeneous Poissonian pair correlations, proving several properties that show similarities and differences to its homogeneous counterpart. In particular, we show that sequences with inhomogeneous Poissonian pair…

We show that the existence of arithmetic progressions with few primes, with a quantitative bound on "few", implies the existence of larger gaps between primes less than x than is currently known unconditionally. In particular, we derive…

We show that a positive proportion of all gaps between consecutive primes are small gaps. We provide several quantitative results, some unconditional and some conditional, in this flavour.

For a sequence of integers $\{a(x)\}_{x \geq 1}$ we show that the distribution of the pair correlations of the fractional parts of $\{ \langle \alpha a(x) \rangle \}_{x \geq 1}$ is asymptotically Poissonian for almost all $\alpha$ if the…

In this paper, we introduce the notion of a ``pairwise independent correlation gap'' for set functions with random elements. The pairwise independent correlation gap is defined as the ratio of the maximum expected value of a set function…

The Pursley-Sarwate criterion of a pair of finite complex-valued sequences measures the collective smallness of the aperiodic autocorrelations and the aperiodic crosscorrelations of the two sequences. It is known that this quantity is…

The most common difference that occurs among the consecutive primes less than or equal to $x$ is called a jumping champion. Occasionally there are ties. Therefore there can be more than one jumping champion for a given $x$. In 1999 A.…

M. Levin defined a real number $x$ that satisfies that the sequence of the fractional parts of $(2^n x)_{n\geq 1}$ are such that the first $N$ terms have discrepancy $O((\log N)^2/ N)$, which is the smallest discrepancy known for this kind…