Iterates of functions defined in terms of digital representations of the integers

Thiel, Johann A.

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https://hdl.handle.net/2142/26109 Copy

Description

Title

Iterates of functions defined in terms of digital representations of the integers

Author(s)

Thiel, Johann A.

Issue Date

2011-08-25T22:14:28Z

Director of Research (if dissertation) or Advisor (if thesis)

Hildebrand, A.J.

Doctoral Committee Chair(s)

Berndt, Bruce C.

Committee Member(s)

• Hildebrand, A.J.
• Reznick, Bruce A.
• Stolarsky, Kenneth B.

Department of Study

Mathematics

Discipline

Mathematics

Degree Granting Institution

University of Illinois at Urbana-Champaign

Degree Name

Ph.D.

Degree Level

Dissertation

Keyword(s)

• Conway's RATS
• iterative process
• Lehmer
• palindromes
• problem 196
• discrete dynamical systems
• quasiperiodic
• Erd ̋os-Kac
• base
• Lyndon words
• Reverse-Add-Then-Sort (RATS)

Abstract

For a fixed base, John H. Conway’s RATS sequences are generated by iterating the following procedure on an initial integer: Reverse the digits of the integer, Add the reversal to the original, Then Sort the resulting digits in increasing order. For example, 334+433=767, which gets sorted into 677. In base 10, Conway discovered the curious sequence: 12333334444, 55666667777, 123333334444, 556666667777, .... Although the sequence is not periodic, it does display some periodic-like behavior which we refer to as “quasiperiodic.” Conway conjectured that all RATS sequences in base 10 are either eventually periodic, or they eventually lead to the previously mentioned quasiperiodic sequence. In this thesis, we study RATS sequences in various bases. In particular, we prove an Erd ̋os-Kac type result for the periods of RATS sequences in base 3; we establish a connection between RATS sequences in general bases and Lyndon words; and we construct infinite families of bases for which there exist RATS sequences having certain prescribed periodicity properties, e.g., we show that there are infinitely many bases for which we can construct quasiperiodic RATS sequences all of a similar type. In the final chapter, we consider a similar iteration process, the reverse-add process. We present data and heuristic arguments on a problem of D.H. Lehmer asking whether every sequence obtained by this process contains a palindrome.

Graduation Semester

2011-08

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http://hdl.handle.net/2142/26109

Copyright and License Information

Copyright 2011 Johann A. Thiel

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