Related papers: Expansions in non-integer bases: lower order revis…

Let $\beta\in(1,2)$. Each $x\in[0,\frac{1}{\beta-1}]$ can be represented in the form \[ x=\sum_{k=1}^\infty \epsilon_k\beta^{-k}, \] where $\epsilon_k\in\{0,1\}$ for all $k$ (a $\beta$-expansion of $x$). If $\beta>\frac{1+\sqrt5}{2}$, then,…

We discuss three convolution inequalities that are connected to additive combinatorics. Cloninger and the second author showed that for nonnegative $f \in L^1(-1/4, 1/4)$, $$ \max_{-1/2 \leq t \leq 1/2} \int_{\mathbb{R}}{f(t-x) f(x) dx}…

In this paper, our main focus is expressing real numbers on the non-integer bases. We denote those bases as $\beta$'s, which is also a real number and $\beta \in (1,2)$. This project has 3 main parts. The study of expansions of real numbers…

We prove that finite sets of real numbers satisfying $|AA| \leq |A|^{1+\epsilon}$ with sufficiently small $\epsilon > 0$ cannot have small additive bases nor can they be written as a set of sums $B+C$ with $|B|, |C| \geq 2$. The result can…

The bivariate series $\theta (q,x):=\sum _{j=0}^{\infty}q^{j(j+1)/2}x^j$ defines a {\em partial theta function}. For fixed $q$ ($|q|<1$), $\theta (q,.)$ is an entire function. We prove a property of stabilization of the coefficients of the…

We obtain exact and asymptotic expressions for the excess of nonnegative (2m+1)-multiples less than (2m)^k with even digit sums in the base 2m.

In this paper, we prove that the discrete Copson inequality (E.T. Copson, \emph{Notes on a series of positive terms}, J. London Math. Soc., 2 (1927), 49-51) of one-dimension in general cases admits an improvement. In fact we study the…

For two real bases $q_0, q_1 > 1$, a binary sequence $i_1 i_2 \cdots \in \{0,1\}^\infty$ is the $(q_0,q_1)$-expansion of the number \[ \pi_{q_0,q_1}(i_1 i_2 \cdots) = \sum_{k=1}^\infty \frac{i_k}{q_{i_1} \cdots q_{i_k}}. \] Let…

For k>=3 let A \subset [1,N] be a set not containing a solution to a_1 x_1+...+a_k x_k=a_1 x_{k+1}+...+a_k x_{2k} in distinct integers. We prove that there is an epsilon>0 depending on the coefficients of the equation such that every such A…

Let $\mathbb{N}$ be the set of all nonnegative integers. For any positive integer $k$ and any subset $A$ of nonnegative integers, let $r_{1,k}(A,n)$ be the number of solutions $(a_1,a_2)$ to the equation $n=a_1+ka_2$. In 2016, Qu proved…

Let $D,Q$ be natural numbers, $(D,Q)=1$, such that $D/Q>1$ and $D/Q$ is not a square. Let $q$ be the smallest divisor of $Q$ such that $Q|\, q^2$. We show that the units $>1$ of the ring $\mathbb Z[\sqrt{Dq^2/Q}]$ are connected with certain…

A sequence $\{x_{n}\}_1^\infty$ in $[0,1)$ is called Borel-Cantelli (BC) if for all non-increasing sequences of positive real numbers $\{a_n\}$ with $\underset{i=1}{\overset{\infty}{\sum}}a_i=\infty$ the set…

An $\mathbb{F}_q$-linear set of rank $k$ on a projective line $\mathrm{PG}(1,q^h)$, containing at least one point of weight one, has size at least $q^{k-1}+1$ (see [J. De Beule and G. Van De Voorde, The minimum size of a linear set, J.…

Let $n$ be a positive integer. Denote by $\mathrm{PG}(n,q)$ the $n$-dimensional projective space over the finite field $\mathbb{F}_q$ of order $q$. A blocking set in $\mathrm{PG}(n,q)$ is a set of points that has non-empty intersection with…

Let $U_q$ be the quantum group corresponding to a complex simple Lie algebra $\mathfrak g$ with root system $R$. Assume the quantum parameter $q\in \C$ is a root of unity. In this paper we study the extensions between simple modules in the…

A partial $(k-1)$-spread in $\operatorname{PG}(n-1,q)$ is a collection of $(k-1)$-dimensional subspaces with trivial intersection, i.e., each point is covered at most once. So far the maximum size of a partial $(k-1)$-spread in…

Let $\mathbb{F}_{q}$ be a finite field with $q$ elements and $\mathbb{F}_{q}[x]$ the ring of polynomials over $\mathbb{F}_{q}$. Let $l(x), k(x)$ be coprime polynomials in $\mathbb{F}_{q}[x]$ and $\Phi(k)$ the Euler function in…

A $(k,\varepsilon)$-non-malleable extractor is a function ${\sf nmExt} : \{0,1\}^n \times \{0,1\}^d \to \{0,1\}$ that takes two inputs, a weak source $X \sim \{0,1\}^n$ of min-entropy $k$ and an independent uniform seed $s \in \{0,1\}^d$,…

Uniform asymptotic expansions are derived for the zeros of the reverse generalized Bessel polynomials of large degree $n$ and real parameter $a$. It is assumed that $-\Delta_{1} n+\frac{3}{2} \leq a \leq \Delta_{2} n$ for fixed arbitrary…

Let $\mathcal{A}$ be a finite subset of $\mathbb{N}$ including $0$ and $f_\mathcal{A}(n)$ be the number of ways to write $n=\sum_{i=0}^{\infty}\epsilon_i2^i$, where $\epsilon_i\in\mathcal{A}$. We consider asymptotics of the summatory…