Related papers: Exceptional points for Lebesgue's density theorem …

A set of points with finite density is constructed in $\mathbb{R}^d$, with $d\geq2$, by adding points to a Poisson process such that any line segment of length $O\left(\varepsilon^{-(d-1)}\ln\varepsilon^{-1}\right)$ in $\mathbb{R}^d$ will…

We study algorithmic randomness notions via effective versions of almost-everywhere theorems from analysis and ergodic theory. The effectivization is in terms of objects described by a computably enumerable set, such as lower semicomputable…

The Bernstein approximation problem is to determine whether or not the space of all polynomials is dense in a given weighted $C_0$-space on the real line. A theorem of L. de Branges characterizes non--density by existence of an entire…

In this note we use recent developments in sieve theory to highlight the interplay between Goldbach and de Polignac numbers. Assuming that the primes have level of distribution greater than $1/2$, we show that at least one of two nice…

We introduce the Density Formula for (topological) drawings of graphs in the plane or on the sphere, which relates the number of edges, vertices, crossings, and sizes of cells in the drawing. We demonstrate its capability by providing…

We are concerned with the existence and uniqueness of solutions with only bounded density for the barotropic compressible Navier-Stokes equations. Assuming that the initial velocity has slightly sub-critical regularity and that the initial…

A.Olevskii and A.Ulanovskii obtained a scale of density results, which correspond to how well an exponential system approximates a uniformly minimal system over a compact set. We extend their result in several directions. First, we show…

Chebotarev's density theorem asserts that the prime ideals are equidistributed among the conjugacy classes of the Galois group of any normal extension of number fields. An effective version of this theorem was first established by Lagarias…

In this short note we discuss the exceptional locus for the Lang-Vojta's conjecture in the case of the complement of two completely reducible hyperplane sections in a cubic surface. Using elementary methods, we show that generically the…

Let $\mathcal{P}$ denote the set of all primes, and let $\underline\delta(P)$ denote the relative lower density of a subset $P$ in $\mathcal{P}$. Suppose that $P_1, P_2, P_3, P_4$ are four subsets of primes with…

Complex moment sequences are exactly those which admit positive definite extensions on the integer lattice points of the upper diagonal half-plane. Here we prove that the aforesaid extension is unique provided the complex moment sequence is…

This paper presents some finite combinatorics of set systems with applications to model theory, particularly the study of dependent theories. There are two main results. First, we give a way of producing lower bounds on VC_ind-density, and…

We consider the complex $\nu$ plane structure of the associated Legendre function of the second kind $Q^{-1/2-K}_{\nu}(\cosh\rho)$. We find that for any noninteger value for $K$ $Q^{-1/2-K}_{\nu}(\cosh\rho)$ has an infinite number of poles…

It has been conjectured that the real parts of the zeros of a linear combination of two or more $L$-functions are dense in the interval $[1,\sigma^*]$, where $\sigma^*$ is the least upper bound of the real parts of such zeros. In this paper…

In this paper, we introduce the notion of the universe, induced communities, and cells with their corresponding spots. Using this language, we formulate and prove the union close set conjecture by showing that for any finite universe…

Let $X \subset \mathbb{P}^n$ be a non-singular hypersurface of degree $d>1$, and let $\epsilon>0$. This paper is concerned with the conjecture that there are $O(B^{n-1+\epsilon})$ rational points on $X$ that have height at most $B$, in…

We prove the density hypothesis for congruence subgroups of an irreducible uniform lattice in $\mathrm{PSL}_2(\mathbb{R})^d$, extending previous results on the spherical density hypothesis to bound multiplicities of non-tempered…

Let X be a d-dimensional diffusion and M the running supremum of its first component. In this paper, we show that for any t > 0, the density (with respect to the d + 1-dimensional Lebesgue measure) of the pair (Mt, Xt) is a weak solution of…

We prove the universality of the large deviations principle for the empirical measures of zeros of random polynomials whose coefficients are i.i.d. random variables possessing a density with respect to the Lebesgue measure on C, R or R + ,…

We prove that any numerically exceptional collection of maximal length, consisting of line bundles, on a smooth del Pezzo surface is a standard augmentation in the sense of L.Hille and M.Perling. We deduce that any such collection is…