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Smooth sums with small spacings

Wouter van Doorn, Anneroos R. F. Everts

TL;DR

The paper addresses the problem of representing every positive integer $n$ as a sum $n=\sum b_i$ of distinct $A_p$-elements with a spacing constraint $b_r < C_p b_1$, extending Erd\"os' work on $3$-smooth numbers. It develops a constructive framework around a finite base set $S\subset A_p\setminus\{1\}$ and layered sequences $(M_k)$, $(u_k)$, $(P_k)$ to perform a midgame transformation that converts a general representation into a $0/1$-coefficient form, yielding the bound $C_p<\tfrac{1}{2}F(4p)$. Special cases when $p-1$ or $p+1$ is a power of two give explicit constants $C_p=2p$ or $C_p=2(p+1)$, and a lower bound $C_p>p$ shows the spacing cannot be arbitrarily small. The authors also discuss multiset-based refinements that can further shrink $C_p$ in certain instances, highlighting open challenges for a universal linear bound in $p$.

Abstract

Solving a problem by Erdős, we prove that every positive integer $n$ can be written as a sum $$n = b_{1} + b_{2} + \ldots + b_{r}$$ of distinct $3$-smooth integers with $1 \le b_{1} < b_{2} < \ldots < b_{r} < 6b_{1}$.

Smooth sums with small spacings

TL;DR

The paper addresses the problem of representing every positive integer as a sum of distinct -elements with a spacing constraint , extending Erd\"os' work on -smooth numbers. It develops a constructive framework around a finite base set and layered sequences , , to perform a midgame transformation that converts a general representation into a -coefficient form, yielding the bound . Special cases when or is a power of two give explicit constants or , and a lower bound shows the spacing cannot be arbitrarily small. The authors also discuss multiset-based refinements that can further shrink in certain instances, highlighting open challenges for a universal linear bound in .

Abstract

Solving a problem by Erdős, we prove that every positive integer can be written as a sum of distinct -smooth integers with .

Paper Structure

This paper contains 3 sections, 4 theorems, 14 equations.

Table of Contents

  1. Introduction
  2. Main result
  3. A final optimization

Key Result

Theorem 1

For every odd integer $p > 1$ a constant $C_p$ exists such that every positive integer $n$ can be written as a sum with $b_i \in A_p$ for all $i$ and $b_{1} < b_{2} < \ldots < b_{r} < C_pb_1$. In general the constant $C_p$ can be taken to be equal to $\frac{1}{2}F(4p)$, and if either $p-1$ or $p+1$ is a power of two, then one can take $C_p = 2p$ or $C_p = 2(p+1)$ respectively. On the other hand,

Theorems & Definitions (7)

  • Theorem
  • proof
  • Lemma 1
  • Lemma 2
  • proof
  • Lemma 3
  • proof