Related papers: A uniform set with fewer than expected arithmetic …

By optimal fluctuation method, we study short-time distribution $P(\mathcal{A}=A)$ of the functionals, $\mathcal{A}=\int_{0}^{t_f} x^n(t) dt$, along constrained trajectories of random acceleration process for a given time duration $t_f$,…

Let $A$ be a subset of a finite abelian group such that $A$ has a small difference set $A-A$ and the density of $A$ is small. We prove that, counter--intuitively, the smallness (in terms of $|A-A|$) of the Fourier coefficients of $A$…

In a previous paper of the authors, we showed that for any polynomials $P_1,\dots,P_k \in \Z[\mathbf{m}]$ with $P_1(0)=\dots=P_k(0)$ and any subset $A$ of the primes in $[N] = \{1,\dots,N\}$ of relative density at least $\delta>0$, one can…

In its usual form, Freiman's 3k-4 theorem states that if A and B are subsets of the integers of size k with small sumset (of size close to 2k) then they are very close to arithmetic progressions. Our aim in this paper is to strengthen this…

Numerical characteristics of identities of finite-dimensional nonassociative algebras are studied. The main result is the construction of a four-dimensional simple unitary algebra with fractional PI-exponent strictly less than its…

We describe the structure of a set of integers $A$ of positive density $\delta$, such that $A+A$ contains no squarefree integer. It turns out that the behaviour changes abruptly at the values $\delta_0=1/4-\frac{2}{\pi^2}=0.0473...$ and…

We show that for some constant $\beta > 0$, any subset $A$ of integers $\{1,\ldots,N\}$ of size at least $2^{-O((\log N)^\beta)} \cdot N$ contains a non-trivial three-term arithmetic progression. Previously, three-term arithmetic…

Let $A$ be a random subset of $\mathbb{Z}_{N}$ obtained by including each element of $\mathbb{Z}_{N}$ in $A$ independently with probability $p$. We say that $A$ is \emph{linear} if the only Freiman homomorphisms are given by the…

A set ${\cal A} \subseteq \Set{1,...,N}$ is of type $B_2$ if all sums $a+b$, with $a\ge b$, $a,b\in {\cal A}$, are distinct. It is well known that the largest such set is of size asymptotic to $N^{1/2}$. For a $B_2$ set ${\cal A}$ of this…

Permutations of the positive integers avoiding arithmetic progressions of length $5$ were constructed in (Davis et al, 1977), implying the existence of permutations of the integers avoiding arithmetic progressions of length $7$. We…

The problem of looking for subsets of the natural numbers which contain no 3-term arithmetic progressions has a rich history. Roth's theorem famously shows that any such subset cannot have positive upper density. In contrast, Rankin in 1960…

We give asymptotic large deviations estimates for the volume inside a domain U of the zero set of a random polynomial of degree N, or more generally, of a holomorphic section of the N-th power of a positive line bundle on a compact Kaehler…

In this paper we investigate the uniform distribution properties of polynomials in many variables and bounded degree over a fixed finite field F of prime order. Our main result is that a polynomial P : F^n -> F is poorly-distributed only if…

Given a pair of distinct non-CM normalized eigenforms having integer Fourier coefficients $a_1 (n)$ and $a_2(n)$, we count positive integers $n$ with $(a_1(n), a_2(n))=1$ and make a conjecture about the density of the set of primes $p$ for…

A strictly increasing sequence of positive integers is called a slightly curved sequence with small error if the sequence can be well-approximated by a function whose second derivative goes to zero faster than or equal to $1/x^\alpha$ for…

We establish the following quantitative form of the Green--Tao theorem: if a set $\mathcal{A}$ of relative density $\delta$ within the primes up to $N$ contains no nontrivial arithmetic progressions of length $k\geq 4$, then $\delta\ll…

Let $F\in\mathbb{Z}[x,y]$ be some polynomial of degree 2. In this paper we find the asymptotic behaviour of the least common multiple of the values of $F$ up to $N$. More precisely, we consider $\psi_F(N) =…

The asymptotic probability density function of nonlinear phase noise, often called the Gordon-Mollenauer effect, is derived analytically when the number of fiber spans is very large. The nonlinear phase noise is the summation of infinitely…

Let a be a positive integer which is not a perfect h-th power with h greater than 1, and Q_a(x;4,j) be the set of primes p less than x such that the residual order of a(mod p) is congruent to j modulo 4. When j=0, 2, it is known that…

Let a be a positive integer greater than 1, and Q_a(x;k,j) be the set of primes p less than x such that the residual order of a(mod p) is congruent to j modulo k. In this paper, the natural densities of Q_a(x;4,j) (j=0,1,2,3) are…