Related papers: Distribution properties of compressing sequences d…

Let $(\mathcal{A}_i)_{i \in [s]}$ be a sequence of dense subsets of the Boolean cube $\{0,1\}^n$ and let $p$ be a prime. We show that if $s$ is assumed to be superpolynomial in $n$ then we can find distinct $i,j$ such that the two…

We consider limit probabilities of first order properties in random graphs with a given degree sequence. Under mild conditions on the degree sequence, we show that the closure set of limit probabilities is a finite union of closed…

For nonstationary, strongly mixing sequences of random variables taking their values in a finite-dimensional Euclidean space, with the partial sums being normalized via matrix multiplication, with certain standard conditions being met, the…

In this second part of our survey on the social and natural distributions, we investigate some models, which intend to explain the statistical regularity of the natural and social distributions. There is a large variety of models and in…

We adopt a physically motivated empirical approach to the characterisation of the distributions of twin and triplet primes within the set of primes, rather than in the set of all natural numbers. Remarkably, the occurrences of twins or…

We examine the extent to which sublinear-sample property testing and estimation apply to settings where samples are independently but not identically distributed. Specifically, we consider the following distributional property testing…

We give a large sieve type inequality for functions supported on primes. As application we prove a conjecture by Elliott, and give bounds for short character sums over primes. The proves uses a combination of the large sieve and the Selberg…

In recent work, we considered the frequencies of patterns of consecutive primes $\pmod{q}$ and numerically found biases toward certain patterns and against others. We made a conjecture explaining these biases, the dominant factor in which…

We consider the specification of prior distributions for Bayesian model comparison, focusing on regression-type models. We propose a particular joint specification of the prior distribution across models so that sensitivity of posterior…

We encode the sequence of prime numbers into simple superpositions of identical waves, mimicking the archetypal prime number sieve of Eratosthenes. The primes are identified as zeros accompanied by phase singularities in a physically…

In this paper we study some sophisticated supercongruences involving dual sequences. For $n=0,1,2,\ldots$ define $$d_n(x)=\sum_{k=0}^n\binom nk\binom xk2^k$$ and $$s_n(x)=\sum_{k=0}^n\binom nk\binom xk\binom{x+k}k=\sum_{k=0}^n\binom…

We study the divided power structures over a product of operads with distributive law. We give a systematic method to characterise the divided power algebras over such a product from the structures of divided power algebra coming from each…

The prime numbers have been a source of fascination for millenia and continue to surprise us. Motivated by the hyperuniformity concept, which has attracted recent attention in physics and materials science, we show that the prime numbers in…

Let the group $G = AB$ be the product of the subgroups $A$ and $B$. We determine some structural properties of $G$ when the $p$-elements in $A\cup B$ have prime power indices in $G$, for some prime $p$. More generally, we also consider the…

We show that the sequence of integers which have nearly the typical number of distinct prime factors forms a Poisson process. More precisely, for $\de$ arbitrarily small and positive, the nearest neighbor spacings between integers $n$ with…

Sums over inverse s-th powers of semiprimes and k-almost primes are transformed into sums over products of powers of ordinary prime zeta functions. Multinomial coefficients known from the cycle decomposition of permutation groups play the…

This text provides very easy and short proofs of some basic properties of complex power series (addition, subtraction, multiplication, division, rearrangement, composition, differentiation, uniqueness, Taylor's series, Principle of…

We suggest other models of sieve generated sequences like the Sieve of Eratosthenes to explain randomness properties of the prime numbers, like the twin prime conjecture, the lim sup conjecture, the Riemann conjecture, and the prime number…

English words and the outputs of many other natural processes are well-known to follow a Zipf distribution. Yet this thoroughly-established property has never been shown to help compress or predict these important processes. We show that…

We provide an ergodic theory framework to study statistical properties of smooth sequences over the odd alphabet {1, 3}. The arithmetic nature of this alphabet yields a partition of the subshift of smooth sequences based on their local…