Related papers: Alternating geometric progressions modulo one and …

In this paper, in addition to the earlier introduced involutive divisions, we consider a new class of divisions induced by admissible monomial orderings. We prove that these divisions are noetherian and constructive. Thereby each of them…

I show that a trivial modification of a standard proof of the Roth's Theorem on triples in arithmetic progression would lead to the following Theorem: If A is a "large set" that is its elements are monotone increasing integers and the sum…

In this paper we give a generalization of a linear algebra estimate that occurs in the paper \cite{RS}, by Michael Rosen and Joseph H. Silverman. In \cite{RS} authors give a bound for the size of a submodule of $(\mathbb{Z}/n \mathbb{Z})^2$…

In this paper, we study $k$-term arithmetic progressions $N, N+d, ..., N+(k-1)d$ of powerful numbers. Under the $abc$-conjecture, we obtain $d \gg_\epsilon N^{1/2 - \epsilon}$. On the other hand, there exist infinitely many $3$-term…

We prove that if one has k non-intersecting arithmetic progressions of integers, with common differences 2 <= q_1,...,q_k <= x, then k < x exp((-1/6 + o(1)) sqrt(log x loglog x)). This improves a result of Szemeredi and Erdos.

We obtain for the Kempner series (i.e. harmonic series where certain digits are excluded from all denominators, for example the digit 9 in base 10) new representations as geometrically convergent series. The coefficients for these…

Despite considerable progress in image classification tasks, classification models seem unaffected by the images that significantly deviate from those that appear natural to human eyes. Specifically, while human perception can easily…

Let $\Gamma$ be an infinite discrete subgroup of Gl$_n(\mathbb{C})$. Then either $(\mathbb{R}, <, +, \cdot, \Gamma)$ is interdefinable with $(\mathbb{R}, <, +, \cdot, \lambda^\mathbb{Z})$ for some $\lambda \in \mathbb{R}$, or $(\mathbb{R},…

We show that alternating Turing machines, with a novel and natural definition of acceptance, accept precisely the inductive (Pi-1-1) languages. Total alternating machines, that either accept or reject each input, accept precisely the…

Prunescu and Sauras-Altuzarra showed that all C-recursive sequences of natural numbers have an arithmetic div-mod representation that can be derived from their generating function. This representation consists of computing the quotient of…

We show how to use extended word series in the reduction of continuous and discrete dynamical systems to normal form and in the computation of formal invariants of motion in Hamiltonian systems. The manipulations required involve complex…

We relate binary words with a given number of subsequences to continued fractions of rational numbers with a given denominator. We deduce that there are binary strings of length $O(\log n \log \log n)$ with exactly $n$ subsequences; this…

For $s=3,4$, we prove the existence of arbitrarily long sequences of consecutive integers none of which is a sum of $s$ nonnegative $s$-th powers. More generally, we study the existence of gaps between the values $\leq N$ of diagonal forms…

The alternating (zigzag) numbers $A_n$, counting the ascending alternating permutations of $\left\{1,\cdots,n\right\}$ and defined by the exponential generating function $\tan x+\sec x$, admit several classical combinatorial and analytic…

An abelian square is the concatenation of two words that are anagrams of one another. A word of length $n$ can contain at most $\Theta(n^2)$ distinct factors, and there exist words of length $n$ containing $\Theta(n^2)$ distinct…

We define Collatz representations for a subset of rational numbers and prove that each real number \( x \notin (-1,1) \) can be approximated arbitrarily well by rational numbers which have only \( 2 \)'s and \( 1 \)'s in their Collatz…

Let $x$ be a periodic continued fraction with the initial block $0$ and the repeating block $c_1,\ldots,c_n$. So $x$ is a quadratic irrational of the form $x=a+\sqrt b$, where $a$, $b$ are rational numbers, $b>0$, $b$ not a square. The…

Central, standard, and Christoffel words are three strongly interrelated classes of binary finite words which represent a finite counterpart of characteristic Sturmian words. A natural arithmetization of the theory is obtained by…

In order to prove irrationality of \sqrt{2} by using only decimal expansions (and not fractions), we develop in detail a model of real numbers based on infinite decimals and arithmetic operations with them.

Using a method of H. Davenport and W. M. Schmidt, we show that, for each positive integer n, the ratio 2/n is the optimal exponent of simultaneous approximation to real irrational numbers 1) by all conjugates of algebraic numbers of degree…