Related papers: On a set-theoretic invariant

A variation on the sum-product problem seeks to show that a set which is defined by additive and multiplicative operations will always be large. In this paper, we prove new results of this type. In particular, we show that for any finite…

If the equation of the title has an integer solution with k>=2, then m>10^{10^6}. Leo Moser showed this in 1953 by amazingly elementary methods. With the hindsight of more than 50 years his proof can be somewhat simplified. We give a…

Let $a$ and $b$ be relatively prime positive integers. In this paper the weighted sum $\sum_{n\in{\rm NR}(a,b)}\lambda^{n-1}n^m$ is given explicitly or in terms of the Apostol-Bernoulli numbers, where $m$ is a nonnegative integer, and ${\rm…

We prove that the set of (r_1,r_2,..,r_{d})-badly approximable vectors is a winning set if r_1=r_2=...=r_{d-1}\geq r_{d}.

Recently, Glasby and Paseman considered the following sequence of binomial sums $\{2^{-r}\sum_{i=0}^{r}\binom{m}{i}\}_{r=0}^{m}$ and showed that this sequence is unimodal and attains its maximum value at $r=\lfloor\frac{m}{3}\rfloor+1$ for…

We show that for each n-tuple of positive rational integers (a_1,..,a_n) there are sets of primes S of arbitrarily large cardinality s such that the solutions of the equation a_1x_1+...+a_nx_n=1 with the x_i all S-units are not contained in…

For relatively prime positive integers $u_0$ and $r$ and for $0\le k\le n$, define $u_k:=u_0+kr$. Let $L_n:={\rm lcm}(u_0, u_1, ..., u_n)$ and let $a, l\ge 2$ be any integers. In this paper, we show that, for integers $\alpha \geq a$ and…

We give a short and elementary argument answering a question of Erd\H{o}s and Ingham negatively. Erd\H{o}s and Ingham showed that a Tauberian estimate they considered was equivalent to the non-vanishing of $1+\sum_{k}a_k^{-1-it}$ for any…

Linear sets on the projective line have attracted a lot of attention because of their link with blocking sets, KM-arcs and rank-metric codes. In this paper, we study linear sets having two points of complementary weight, that is with two…

The main objective of this paper is to investigate the explicit form, stability character and global behavior of solutions of the following two systems of rational difference equations x_{n+1}=((+(-)1)/(y_{n}(x_{n-1}+(-)1)+1)),…

We consider a variant of the ABC Conjecture, attempting to count the number of solutions to $A+B+C=0$, in relatively prime integers $A,B,C$ each of absolute value less than $N$ with $r(A)<|A|^a, r(B)<|B|^b, r(C)<|C|^c.$ The ABC Conjecture…

The aim of this paper is threefold: a) Finding new direct and inverse results in the additive number theory concerning Minkowski sums of dilates. b) Finding a connection between the above results and some direct and inverse problems in the…

A new class of integrable mappings and chains is introduced. Corresponding $(1+2)$ integrable systems invariant with respect to such discrete transformations are presented in an explicit form. Their soliton-type solutions are constructed in…

For any pair of real numbers $(i,j)$ with $0<i,j<1$ and $i+j=1$, we prove that the set of $p$-adic mixed $(i,j)$-badly approximable numbers $\bad_p(i, j)$ is 1/2-winning in the sense of Schmidt's game. This improves a recent result of…

Given a set of $n$ positive integers $\{a_1, \ldots, a_n\}$ and an integer parameter $H$ we study small additive shifts of its elements by integers $h_i$ with $|h_i| \le H$, $i =1, \ldots, n$, such that the greatest common divisor of…

We compare a piecewise linear map with constant slope beta>1 and a piecewise linear map with constant slope -beta. These maps are called the positive and negative beta-transformations. We show that for a certain set of beta's, the…

We consider a translation invariant linear equation in four variables with integer coefficients of the form: $ax_1 +bx_2= cy_1+dy_2$. The main result of the paper states that any set on the real line with Fourier dimension greater than 1/2…

In this paper we resolve a conjecture of Kresch and Tamvakis. Our result is the following. Theorem: For any positive integer $D$ and any integers $i,j$ $(0\leq i,j \leq D)$, the absolute value of the following hypergeometric series is at…

In this paper, we study the weighted difference substitutions from geometrical views. First, we give the geometric meanings of the weighted difference substitutions, and introduce the concept of convergence of the sequence of substitution…

In this paper we study recurrences concerning the combinatorial sum $[n,r]_m=\sum_{k\equiv r (mod m)}\binom {n}{k}$ and the alternate sum $\sum_{k\equiv r (mod m)}(-1)^{(k-r)/m}\binom{n}{k}$, where m>0, $n\ge 0$ and r are integers. For…