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Common substring with shifts in b-ary expansions

Xin Liao, Dingding Yu

Abstract

Denote by $S_n(x,y)$ the length of the longest common substring of $x$ and $y$ with shifts in their first $n$ digits of $b$-ary expansions. We show that the sets of pairs $(x,y)$, for which the growth rate of $S_n(x,y)$ is $α\log n$ with $0\le α\le \infty$, have full Hausdorff dimension.

Common substring with shifts in b-ary expansions

Abstract

Denote by the length of the longest common substring of and with shifts in their first digits of -ary expansions. We show that the sets of pairs , for which the growth rate of is with , have full Hausdorff dimension.
Paper Structure (4 sections, 7 theorems, 47 equations)

This paper contains 4 sections, 7 theorems, 47 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of Theorem \ref{['main theorem1']}
  4. Proof of Theorem \ref{['main theorem3']}

Key Result

Theorem 1.1

Let $S(\alpha)$ be defined as in (S alpha). Then

Theorems & Definitions (13)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • proof : Proof of Theorem \ref{['main theorem1']}
  • ...and 3 more