Avoiding large squares in infinite binary words

  • Authors:

  • Narad Rampersad;Jeffrey Shallit;Ming-wei Wang

  • Affiliations:

  • School of Computer Science, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada;School of Computer Science, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada;School of Computer Science, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada

  • Venue:
  • Theoretical Computer Science - Combinatorics on words
  • Year:
  • 2005

Quantified Score

Hi-index 0.01

Visualization

Abstract

We consider three aspects of avoiding large squares in infinite binary words. First, we construct an infinite binary word avoiding both cubes xxx and squares yy with |y| ≥4; our construction is somewhat simpler than the original construction of Dekking. Second, we construct an infinite binary word avoiding all squares except 02, 12, and (01)2; our construction is somewhat simpler than the original construction of Fraenkel and Simpson. In both cases, we also show how to modify our construction to obtain exponentially many words of length n with the given avoidance properties. Finally, we answer an open question of Prodinger and Urbanek from 1979 by demonstrating the existence of two infinite binary words, each avoiding arbitrarily large squares, such that their perfect shuffle has arbitrarily large squares.