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Clemm and Trebat-Leder (2014) proved that the number of quadratic number fields with absolute discriminant bounded by $x$ over which there exist elliptic curves with good reduction everywhere and rational $j$-invariant is $\gg…

We establish a generalization of Bourgain double recurrence theorem by proving that for any map $T$ acting on a probability space $(X,\mathcal{A},\mu)$, and for any non-constant polynomials $P, Q$ mapping natural numbers to themselves, for…

Let $\gcd(m,n)$ denote the greatest common divisor of the positive integers $m$ and $n$, and let $\mu$ represent the M\" obius function. For any real number $x>5$, we define the summatory function of the M\" obius function involving the…

Let $\kl_n(a,b;m)$ be the hyper-Kloosterman sum. Fix integers $n\geqslant2,a\neq0$, $b\neq0$ and $k\geqslant2$. For any $0\neq\eta\in\mathbb{C}$ and multiplicative function $f: \mathbb{N} \rightarrow \mathbb{C}$, we prove that…

Let $A$ be an expanding endomorphism on the torus ${\Bbb T}^d = {\Bbb R}^d /{\Bbb Z}^d$ with its smallest eigenvalue $\lambda >1$. Consider the ergodic system $({\Bbb T}^d, A, \mu)$ where $\mu$ is Haar measure. We prove that the correlation…

A collection of orthonormal bases for a complex dXd Hilbert space is called mutually unbiased (MUB) if for any two vectors v and w from different bases the square of the inner product equals 1/d: |<v,w>| ^{2}=1/d. The MUB problem is to…

We prove the following results solving a problem raised in [Y. Caro, R. Yuster, On zero-sum and almost zero-sum subgraphs over $\mathbb{Z}$, Graphs Combin. 32 (2016), 49--63]. For a positive integer $m\geq 2$, $m\neq 4$, there are…

A set $B$ is said to be \emph{sum-free} if there are no $x,y,z\in B$ with $x+y=z$. We show that there exists a constant $c>0$ such that any set $A$ of $n$ integers contains a sum-free subset $A'$ of size $|A'|\geqslant n/3+c\log \log n$.…

We state several equivalent noncommutative versions of the Cauchy-Riemann equations and characterize the unbounded operators on L^2(R) which satisfy them. These operators arise from the creation operator via a functional calculus involving…

In this paper, we consider the non-spectral problem for the planar self-affine measures $\mu_{M,D}$ generated by an expanding integer matrix $M\in M_2(\mathbb{Z})$ and a finite digit set $D\subset\mathbb{Z}^2$. Let $p\geq2$ be a positive…

We consider the error term of the asymptotic formula for the number of pairs of $k$-free integers up to $x$. Our error term improves results by Heath-Brown, Brandes and Dietmann/Marmon. We then extend our results to $r$-tuples of $k$-free…

A finite set $A$ of integers is square-sum-free if there is no subset of $A$ sums up to a square. In 1986, Erd\H os posed the problem of determining the largest cardinality of a square-sum-free subset of $\{1, ..., n \}$. Answering this…

Let G be a finite abelian group of order n. For a complex valued function f on G, let \fht denote the Fourier transform of f. The uncertainty inequality asserts that if f \neq 0 then |supp(f)| |supp(\fht)| \geq n. Answering a question of…

We revisit several hybrid multiplicative-to-additive type functions from a recent preprint article. These functions, $g(n)$ with Dirichlet generating function (DGF) $\zeta(s)^{-1} (1+P(s))^{-1}$ for $\Re(s) > 1$ where $P(s) = \sum_p p^{-s}$…

Since the theorems of Schur and van der Waerden, numerous partition regularity results have been proved for linear equations, but progress has been scarce for non-linear ones, the hardest case being equations in three variables. We prove…

In 1946, P. Erd\H os and I. Niven proved that there are only finitely many positive integers $n$ for which one or more elementary symmetric functions of $1, 1/2, \ldots , 1/n$ are integers. In this paper we solve this old problem by showing…

A recent result of Downarowicz and Serafin (DS) shows that there exist positive entropy subshifts satisfying the assertion of Sarnak's conjecture. More precisely, it is proved that if $y=(y_n)_{n\ge 1}$ is a bounded sequence with zero…

Let $(X, \mathcal{A},\mu)$ be a probability space and let $T$ be a contraction on $L^2(\mu)$. We provide suitable conditions over sequences $(w_k)$, $(u_k)$ and $(A_k)$ in such a way that the weighted ergodic limit…

We prove that if $f$ is a random completely multiplicative function, conditional $f(p)=1$ for each prime $p \le (\log x)^{2-\epsilon}$, the probability that $\sum_{1\le n \le N}f(n)\ge 0$ for all $N\le x$ is $o(1)$ as $x \rightarrow…

In this paper we classify the nonnegative global minimizers of the functional \[ J_F(u)=\int_\Omega F(|\nabla u|^2)+\lambda^2\chi_{\{u>0\}}, \] where $F$ satisfies some structural conditions and $\chi_D$ is the characteristic function of a…