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Completeness of exponentially increasing sequences

Wouter van Doorn

Abstract

For fixed positive reals $t$ and $α$, consider the sequence $S_t(α) = (s_1, s_2, \ldots, )$ with $s_n = \left \lfloor tα^n \right \rfloor$. In 1964, Graham managed to characterize those pairs $(t, α)$ with $0 < t < 1$ and $1 < α< 2$ for which every large enough integer can be written as the sum of distinct elements of $S_t(α)$. We show that his methods can be applied to deal with many other pairs of $(t, α)$ as well.

Completeness of exponentially increasing sequences

Abstract

For fixed positive reals and , consider the sequence with . In 1964, Graham managed to characterize those pairs with and for which every large enough integer can be written as the sum of distinct elements of . We show that his methods can be applied to deal with many other pairs of as well.
Paper Structure (4 sections, 17 theorems, 20 equations, 2 figures)

This paper contains 4 sections, 17 theorems, 20 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminary lemmas
  3. Main results
  4. Computational possibilities

Key Result

Lemma 1

Assume that a positive integer $m \notin P(S_t(\alpha))$ and a non-negative integer $r$ exist with while $s_n + s_{n+1} \le s_{n+2}$ holds for all $n > r$. Then for all $k \ge 1$.

Figures (2)

  • Figure :
  • Figure :

Theorems & Definitions (32)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Corollary 1
  • Lemma 3: gra, Lemma $2$
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • ...and 22 more