Shatrovskii's construction of thin bases

TL;DR

This paper presents a construction method for thin additive bases of a given order, which are sparse sets of nonnegative integers capable of representing all nonnegative integers through sums of a fixed number of elements.

Contribution

It introduces a novel construction of thin bases of order h based on Shatrovskii's approach, expanding the understanding of sparse additive bases.

Findings

Constructed explicit examples of thin bases of order h.

Proved the sparsity condition for the constructed bases.

Demonstrated that these bases can represent all nonnegative integers.

Abstract

The set A of nonnegative integers is called a basis of order h if every nonnegative integer can be represented as the sum of exactly h not necessarily distinct elements of A. An additive basis A of order h is called thin if there exists c > 0 such that the number of elements of A not exceeding x is less than cx^{1/h} for all x > 0. This paper describes a construction of Shatrovskii of thin bases of order h.