On additive and multiplicative decompositions of sets of integers with restricted prime factors, I. (Smooth numbers)

On smooth sets of integers

This work studies evenly distributed sets of integers — sets whose quantity within each interval is proportional to the size of the interval, up to a bounded additive deviation. Namely, for ρ,∆ ∈ R a set A of integers is (ρ,∆)-smooth if abs(|I| · ρ − |I ∩ A|) < ∆ for any interval I of integers; a set A is ∆-smooth if it is (ρ,∆)-smooth for some real number ρ. The paper introduces the concept of...

Additive and Multiplicative Properties of Beta - Integers

| To each number > 1 there corresponds a discrete countable set of numbers denoted by Z and named set of beta-integers. The set Z is precisely the set of real numbers which are polynomial in when they are written in \basis ", and Z = Zwhen 2 N. We prove here a list of arithmetic properties of Z : addition, multiplication, relation with integers, when is a quadratic Pisot-Vijayaraghavan unit (qu...

Some Remarks on Prime Factors of Integers

Prime factors of consecutive integers

This note contains a new algorithm for computing a function f(k) introduced by Erdős to measure the minimal gap size in the sequence of integers at least one of whose prime factors exceeds k. This algorithm enables us to show that f(k) is not monotone, verifying a conjecture of Ecklund and Eggleton.

Additive and Multiplicative Structures of C ? - Sets

It is known that for an IP? set A in (N,+) and a sequence hxnin=1 in N, there exists a sum subsystem hynin=1 of hxnin=1 such that FS(hynin=1) [ FP (hynin=1) ✓ A. Similar types of results have also been proved for central? sets where the sequences have been considered from the class of minimal sequences. In this present work, we shall prove some analogous results for C?-sets for a more general c...