Related papers: Problems in additive number theory, I

This paper extends previous work on linear correlations of representation functions of positive definite binary quadratic forms to allow indefinite forms.

We study the concept of universal sets from the additive--combinatorial point of view. Among other results we obtain some applications of this type of uniformity to sets avoiding solutions to linear equations, and get an optimal upper bound…

We present an auxiliary space theory that provides a unified framework for analyzing various iterative methods for solving linear systems that may be semidefinite. By interpreting a given iterative method for the original system as an…

A classical additive basis question is Waring's problem. It has been extended to integer polynomial and non-integer power sequences. In this paper, we will consider a wider class of functions, namely functions from a Hardy field, and show…

The Lie algebras over the algebra of dual numbers are introduced and investigated.

We introduce the concept of a triangular representation of a Lie algebra, give a counterpart of Ado's theorem, and discuss $2$-irreducible triangular modules over a nonreductive Lie algebra.

We connect the well-known theory of functional forms of variational bicomplex with the theory of antiexact differential forms. We identify antiexact functional forms as an obstruction to the variationality of differential equations. The…

Besides various asymptotic results on the concept of sum-product bases in $\mathbb{N}_0$, we consider by probabilistic arguments the existence of thin sets $A,A'$ of integers such that $AA+A=\mathbb{N}_0$ and $A'A'+A'A'=\mathbb{N}_0$.

The additive square problem is a relatively famous open problem in the area of combinatorics on words: Does there exist an infinite word over a finite alphabet, such that no two consecutive blocks of the same length have the same sum? In…

In the Frobenius problem we are given a set of coprime, positive integers $a_1, a_2,...,a_k$, and are interested in the set of positive numbers NR that have no representation by the linear form $\sum_i a_ix_i$ in nonnegative integers $x_1,…

The Waring problem of forms concerns the expression of homogeneous multivariate polynomials as sums of powers of linear forms. This paper focuses on complex binary forms, and we solve the Waring problem for them using basic tools in algebra…

The inverse problem for representation functions takes as input a triple (X,f,L), where X is a countable semigroup, f : X --> N_0 \cup {\infty} a function, L : a_1 x_1 + ... + a_h x_h an X-linear form and asks for a subset A \subseteq X…

Some class of sums which naturally include the sums of powers of integers is considered. A number of conjectures concerning a representation of these sums is made.

We present distributions of countable models and correspondent structural characteristics of complete theories with continuum many types: for prime models over finite sets relative to Rudin-Keisler preorders, for limit models over types and…

Work in progress concerning alternative formalizations of arithmetic.

We address the "sums of dilates problem" by looking for non-trivial lower bounds on sumsets of the form $k \cdot X + l \cdot X$, where $k$ and $l$ are non-zero integers and $X$ is a subset of a possibly non-abelian group $G$ (written…

In this paper we consider the notions of binomial thinning, binomial mixing, their generalizations, certain interplay between them, associated limit theorems and provide various examples.

We introduce a new framework called linear algebraic number theory (LANT) that reformulates the number-theoretic problem as a regression model and solves it using matrix algebra. This framework restricts all computations to log space,…

For a set $A$, let $P(A)$ be the set of all finite subset sums of $A$. In this paper, for a sequence of integers $B=\{1<b_1<b_2<\cdots\}$ and $3b_1+5\leq b_2\leq 6b_1+10$, we determine the critical value for $b_3$ such that there exists an…

This article surveys recent advances and future challenges in the $2$-representation theory of finitary $2$-categories with a particular emphasis on problems related to classification of various classes of $2$-representations.