Related papers: Arithmetic properties of Cantor sets involving non…

The article is devoted to the alternating Cantor series. It is proved that any real number belonging to $[a_0-1;a_0]$, where $a_0=\sum^{\infty} _{k=1} {\frac{d_{2k}-1}{d_1d_2...d_{2k}}} $, has no more than two representations by the series…

Suppose $k$ is a positive integer. In this work, we establish formulas for for the number of representations of integers by the quadratic forms $$ x_{1}^{2}+\cdots+x_{k}^{2}+l\left(x_{k+1}^{2}+\cdots+x_{2k}^{2}\right) $$ for $l\in\{2,4\}$.

Let $a_k(n)$ denotes the number of representations of a non-negative integer $n$ as sum of $k$ quadratic forms of the type $x^2+xy+y^2$ and $a_{\lambda_1,\lambda_2,\lambda_3\dots\lambda_k}(n)$ denotes the number of representations $n$ as a…

For a fixed integer $k \ge 0$, consider representations of positive integers as sums of binomial coefficients of the form $\binom{n}{k}$. While exact minimal bounds for the number of required summands are known only in a few low-dimensional…

Cantor sets of integers have a rich set of arithmetic combinatorial properties. We consider classical Cantor sets, with a base and a fixed set of allowed digits. For such sets, we (a) give examples of such sets that satisfy the intersective…

We investigate here the representability of integers as sums of triangular numbers, where the $n$-th triangular number is given by $T_n = n(n + 1)/2$. In particular, we show that $f(x_1,x_2,..., x_k) = b_1 T_{x_1} +...+ b_k T_{x_k}$, for…

We give formulas for the number of representations of non negative integers by various quadratic forms. We also give evaluations in the case of sum of two cubes (cubic case) and the quintic case, as well. We introduce a class of generalized…

We consider equally-weighted Cantor measures $\mu_{q,b}$ arising from iterated function systems of the form ${b^{-1}(x+i)}$, $i=0,1,...,q-1$, where $q<b$. We classify the $(q,b)$ so that they have infinitely many mutually orthogonal…

For an arbitrary given $k\geq3,$ we consider a possibility of representation of a positive number $n$ by the form $x_1...x_k+x_1+...+x_k, 1\leq x_1\leq ... \leq x_k.$ We also study a question on the smallest value of $k\geq3$ in such a…

We study properties of the set of subsums for a convergent series $ k_1 \sin x + \dots + k_m \sin x +\dots + k_1\sin x^n +\dots + k_m \sin x^n + \dots $, where $k_1, k_2, k_3,\dots,k_m$ are fixed positive integers and $0<x<1$. Depends on…

In the present article, modeling certain rational numbers, that are represented in terms of Cantor series, are described. The statements on relations between digits in the representations of rational numbers by Cantor series (for the case…

Let $C$ be the middle-third Cantor set. In this paper, we show that for every $x\in [0,4]$, there exist $x_1, x_2, x_3, x_4 \in C$ such that $$x= x_1^2+x_2^2+x_3^2+x_4^2,$$ which answers a question posed by Athreya, Reznick,and Tyson.

Let $k\ge 2$ be an integer and let $A$ be a set of nonnegative integers. For a $k$-tuple of positive integers $\underline{\lambda} = (\lambda_{1}, \dots{} ,\lambda_{k})$ with $1 \le \lambda_{1} < \lambda_{2} < \dots{} < \lambda_{k}$, we…

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 give an algorithm that produces all solutions of the equation $\sum_{i=1}^n 1/x_i = 1$ in integers of the form $2^a k^b$, where $k$ is a fixed positive integer that is not a power of $2$, $a$ is an element of $\{0,1,2\}$ that can vary…

In this paper we consider certain quaternary quadratic forms and octonary quadratic forms and by using the theory of modular forms, we find formulae for the number of representations of a positive integer by these quadratic forms.

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…

We prove that a pair of integral quadratic forms in 5 or more variables will simultaneously represent "almost all" pairs of integers that satisfy the necessary local conditions, provided that the forms satisfy a suitable nonsingularity…

Let Q be a non-singular quadratic form with integer coefficients. When Q is indefinite we provide new upper bounds for the least non-trivial integral solution to the equation Q=0. When Q is positive definite we provide improved upper bounds…

Using modular forms we determine formulas for the number of representations of a positive integer by diagonal octonary quadratic forms with coefficients $1$, $2$, $3$ or $6$.