Related papers: Expander estimates for cubes

Let $\mathcal{P}$ denote the set of all primes. $P_{1},P_{2},P_{3}$ are three subsets of $\mathcal{P}$. Let $\underline{\delta}(P_{i})$ $(i=1,2,3)$ denote the lower density of $P_{i}$ in $\mathcal{P}$, respectively. It is proved that if…

In this paper we give a conjectural refinement of the Davenport-Heilbronn theorem on the density of cubic field discriminants. We explain how this refinement is plausible theoretically and agrees very well with computational data.

A central problem in discrete geometry, known as Hadwiger's covering problem, asks what the smallest natural number $N\left(n\right)$ is such that every convex body in ${\mathbb R}^{n}$ can be covered by a union of the interiors of at most…

Here is one of the results obtained in this paper: Let $\Omega\subset {\bf R}^n$ be a smooth bounded domain, let $q>1$, with $q<{{n+2}\over {n-2}}$ if $n\geq 3$ and let $\lambda_1$ be the first eigenvalue of the problem $$\cases{-\Delta…

Starting from an independent-particle model with a finite and arbitrary set of single-particle energies, we develop an analytical approximation to the many-body level density $\rho_A(E)$ and to particle-hole densities. We use exact…

Let A be a pre-defined set of rational numbers. We say a set of natural numbers S is an A-quotient-free set if no ratio of two elements in S belongs to A. We find the maximal asymptotic density and the maximal upper asymptotic density of…

For $p$ being a large prime number, and $A \subset \mathbb{F}_p$ we prove the following: $(i)$ If $A(A+A)$ does not cover all nonzero residues in $\mathbb{F}_p$, then $|A| < p/8 + o(p)$. $(ii)$ If $A$ is both sum-free and satisfies $A =…

It is shown that the density of the values set {Tau(n): n <= x} of the nth coefficients Tau(n) of the discriminant function Delta(z), a cusp form of level N = 1 and weight k = 12, has the lower bound #{Tau(n): n <= x} >> x/log x. The…

It is known that, for each real number x such that 1,x,x^2 are linearly independent over Q, the uniform exponent of simultaneous approximation to (1,x,x^2) by rational numbers is at most (sqrt{5}-1)/2 (approximately 0.618) and that this…

In this article we establish an asymptotic formula for the number of rational points, with bounded denominators, within a given distance to a compact submanifold $\mathcal{M}$ of $\mathbb{R}^M$ with a certain curvature condition. Our result…

Let $s \geq 3$ be a fixed positive integer and $a_1,\dots,a_s \in \mathbb{Z}$ be arbitrary. We show that, on average over $k$, the density of numbers represented by the degree $k$ diagonal form \[ a_1 x_1^k + \cdots + a_s x_s^k \] decays…

We prove a quantitative estimate on the number of certain singularities in almost minimizing clusters. In particular, we consider the singular points belonging to the lowest stratum of the Federer-Almgren stratification (namely, where each…

We show the following result: Assume B is an infinite Boolean Algebra and lambda=d(B). Then s(B*B)$, i.e. s(uf(B)xuf(B))>= lambda$ (if lambda limit - obtained)

We give explicit and asymptotic lower bounds for the quantity $|e^{s/t}-M/N|$ by studying a generalized continued fraction expansion of $e^{s/t}$. In cases $|s|\geq 3$ we improve existing results by extracting a large common factor from the…

In 1969, H. Davenport and W. M. Schmidt studied the problem of approximation to a real number \xi by algebraic integers of degree at most three. They did so, using geometry of numbers, by resorting to the dual problem of finding…

We first construct the global unique solution by assuming that the initial data is small in the $H^3$ norm but the higher order derivatives could be large. If further the initial data belongs to $\Dot{H}^{-s}$ ($0\le s<3/2$) or…

Suppose that A is a subset of the integers {1,...,N} of density a. We provide a new proof of a result of Green which shows that A+A contains an arithmetic progression of length exp(ca(log N)^{1/2}) for some absolute c>0. Furthermore we…

The main result in this paper concerns a new five-variable expander. It is proven that for any finite set of real numbers $A$, $$|\{(a_1+a_2+a_3+a_4)^2+\log a_5 :a_1,a_2,a_3,a_4,a_5 \in A \}| \gg \frac{|A|^2}{\log |A|}.$$ This bound is…

The density bound for schedulability for general pinwheel instances is $\frac{5}{6}$, but density bounds better than $\frac{5}{6}$ can be shown for cases in which the minimum element $m$ of the instance is large. Several recent works have…

The law of large numbers for the empirical density for the pairs of uniformly distributed integers with a given greatest common divisor is a classic result in number theory. In this paper, we study the large deviations of the empirical…