Related papers: On uniformity of $q$-multiplicative sequences

This work is a continuation of [13]. We study the linear disjointness between higher-order oscillating sequences and nonlinear dynamical systems. Specifically, we prove that any oscillating sequence of order $m=d+k-1$ and any simple…

In 2011, Penttila and Williford constructed an infinite new family of primitive $Q$-polynomial 3-class association schemes, not arising from distance regular graphs, by exploring the geometry of the lines of the unitary polar space…

We study Fourier-Bessel series on a q-linear grid, defined as expansions in complete q-orthogonal systems constructed with the third Jackson q-Bessel function, and obtain sufficient conditions for uniform convergence. The convergence…

We obtain new nonexistence results of generalized bent functions from $\{Z^n}_q$ to $\Z_q$ (called type $[n,q]$) in the case that there exist cyclotomic integers in $ \Z[\zeta_{q}]$ with absolute value $q^{\frac{n}{2}}$. This result…

Let $1<q<2$ and \[ \Lambda(q)={\sum_{k=0}^n a_kq^k\mid a_k\in\{-1,0,1\}, n\ge1}. \] It is well known that if $q$ is not a root of a polynomial with coefficients $0,\pm1$, then $\Lambda(q)$ is dense in $\mathbb{R}$. We give several…

We present combinatorial and analytical results concerning a Sheffer sequence with a generating function of the form $G(x,z)=Q(z)^{x}Q(-z)^{1-x}$, where $Q$ is a quadratic polynomial with real zeros. By using the properties of Riordan…

We study the uniform distribution of the polynomial sequence $\lambda(P)=(\lfloor P(k) \rfloor )_{k\geq 1}$ modulo integers, where $P(x)$ is a polynomial with real coefficients. In the nonlinear case, we show that $\lambda(P)$ is uniformly…

We introduce a new family of noncommutative analogues of the Hall-Littlewood symmetric functions. Our construction relies upon Tevlin's bases and simple q-deformations of the classical combinatorial Hopf algebras. We connect our new…

For $q \geq 2$, $n \in \mathbb{N}$, let $s_{q}(n)$ denote the sum of the digits of $n$ written in base $q$. Spiegelhofer (2020) proved that the Thue--Morse sequence has level of distribution $1$, improving on a former result of Fouvry and…

We show that for any positive integer $n$, there exists a quiver $Q$ with $O(n^2)$ vertices and $O(n^2)$ edges such that any quiver on $n$ vertices is a full subquiver of a quiver mutation equivalent to $Q$. We generalize this statement to…

We consider self-similar statistical ensembles with the phase space whose volume is invariant under the deformation that squeezes (expands) the coordinate and expands (squeezes) the momentum. Related probability distribution function is…

A sequence $f_n(q)$ is $q$-holonomic if it satisfies a nontrivial linear recurrence with coefficients polynomials in $q$ and $q^n$. Our main theorems state that $q$-holonomicity is preserved under twisting, i.e., replacing $q$ by $\omega q$…

The Thue--Morse sequence $\{t(n)\}_{n\geqslant 1}$ is the indicator function of the parity of the number of ones in the binary expansion of positive integers $n$, where $t(n)=1$ (resp. $=0$) if the binary expansion of $n$ has an odd (resp.…

We introduce the entangled quantum polynomial hierarchy $\mathsf{QEPH}$ as the class of problems that are efficiently verifiable given alternating quantum proofs that may be entangled with each other. We prove $\mathsf{QEPH}$ collapses to…

Asymptotic expansions are derived for Gegenbauer (ultraspherical) polynomials for large order $n$ that are uniformly valid for unbounded complex values of the argument $z$, including the real interval $0 \leq z \leq 1$ in which the zeros in…

We consider the asymptotics of the correlation functions of the characteristic polynomials of the hermitian Wigner matrices $H_n=n^{-1/2}W_n$. We show that for the correlation function of any even order the asymptotic coincides with this…

In this article, we study the logarithmic Mahler measure of the one-parameter family \[Q_\alpha=y^2+(x^2-\alpha x)y+x,\] denoted by $m(Q_\alpha)$. The zero loci of $Q_\alpha$ generically define elliptic curves $E_\alpha$ which are…

We study the distribution of consecutive sums of two squares in arithmetic progressions. If $\{E_n\}_{n \in \mathbb{N}}$ is the sequence of sums of two squares in increasing order, we show that for any modulus $q$ and any congruence classes…

Fix distinct primes $p$ and $q$ and let $E$ be an elliptic curve defined over a number field $K$. The $(p,q)$-entanglement type of $E$ over $K$ is the isomorphism class of the group $\operatorname{Gal}(K(E[p])\cap K(E[q])/K)$. The size of…

We study the structure of the spectrum of the algebra of uniformly continuous holomorphic functions on the unit ball of $\ell_p$. Our main focus is the relationship between \emph{Gleason parts} and \emph{fibers}. For every $z \in…