Research Statement 1 Behavior of Iterates of Functions
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Research Statement Johann Thiel My general interest is in number theory. In particular, I am interested in analytic number theory and in problems at the interface of number theory and other areas of mathematics. My primary research focus, and the core of my thesis, is in integer sequences generated by iterating functions defined in terms of digital representations of integers. In addition, I have contributed to other problems in number theory that I have encountered in number theory seminar talks and in conversations with faculty members. These include: generalizing a transformation formula for Ramanujan's theta function, supplying a proof for an infinite series identity in Ramanujan's lost notebook, and finding \nice" arrangements of stars on the American flag for any number of stars. 1 Behavior of iterates of functions defined in terms of digital rep- resentations 1.1 Conway's RATS John H. Conway has invented many mathematical games, including the well known Game of Life. In 1990, Conway [2] created a new game he dubbed Reverse-Add-Then-Sort (RATS). The rules are simple: begin with any positive integer n written in base 10, add to it the number obtained by reversing the order of the digits of n, and then sort the result in order of increasing digits from left to right (discarding any zeros). The game stops if the sequence becomes periodic; otherwise, an unbounded sequence is generated. Conway discovered that the RATS sequence generated by starting with 1 eventually leads to the following monotonically increasing sequence ···! 123n34 ! 526n74 ! 123n+134 ! 526n+174 !··· ; where we abbreviate numbers with repeated digits using exponential notation. Conway conjectured that the above example is unique in the following sense: Conjecture 1.1 (Conway [2]) Every RATS sequence is either eventually periodic or eventually part of the above sequence. The RATS game can be generalized for positive integers written in any fixed integer base. For bases smaller than 10, Curtis McMullen [6] conjectured that all RATS sequences are eventually periodic. In base 2, all RATS sequences are trivially periodic. In the case of base 3, the conjecture was verified by Gentges [5]. In all other cases, McMullen's conjecture remains open. Expanding on the base 3 result of Gentges, I found a method for counting integers according to the periodic behavior of the RATS sequence they generate. Theorem 1.2 (Thiel) Let x 2 N and N(x; p) = jfb ≤ x; 2b is eventually periodic with period pgj: Then, for p > 3, (log x)p−1 N(x; p) 2 : (p − 1)! Theorem 1.2 has two interesting consequences. First, it implies that, asymptotically, the distri- bution of periods of RATS sequences generated by 2b, b ≤ x, approximates a Gaussian distribution log2 x centered at 2 . Also, the lemma can be used to bootstrap a similar result for RATS sequence generated by integers of the form 1a2b. McMullen's conjecture for RATS sequences in base 4 and 5 is that they all are eventually periodic. I have performed extensive computations that provide strong support for this conjecture, and in fact suggest that for both bases the eventual period is always 2. In particular, for base 4, I have verified McMullen's conjecture, in the strong form, for all integers ≤ 41000000. Furthermore, I have also investigated the base 4 conjecture theoretically, and have been able to break the analysis down into 15 cases. In all of these cases, with the exception of one, I have been able to prove the conjecture. I hope to complete the remaining case in the near future. In the course of this research, I have discovered a close connection between the behavior of RATS sequences in base 4 and a discrete-time dynamical system somewhat similar to that generated by one of the classic maps in this area, the tent map. I plan to pursue this new angle in the future. Shattuck and Cooper [9] studied the existence of unbounded RATS sequences in various bases. They were able to show that there are infinite families of bases where unbounded RATS sequences can be constructed. Theorem 1.3 (Shattuck, Cooper [9]) Let t be a prime or Fermat pseudoprime in base 2. Then, for any sufficiently large integer m, there exists a positive integer m1 such that t 1 t (t + 1)2 −1 ::: (2t − 1)m (2t)m1 is the start of an unbounded RATS sequence in base (2t − 1)2 + 1. I discovered a surprising connection between their constructions of unbounded sequences and so-called 2-ary Lyndon words, a notion from the theory of combinatorics on words. This connec- tion allowed me to find new infinite families of bases where unbounded RATS sequence can be constructed. Theorem 1.4 (Thiel) For any base of the b = 49n + 1 with n > 0 and any sufficiently large integer m, there exist integers m1; m2; : : : ; m7n+1 such that 1m1 2m2 3m3 ::: (7n)m (7n + 1)m7n+1 is the start of an unbounded RATS sequence in base b. The above theorem extends the result of Theorem 1.3 for t = 3. In fact, I have shown that a version of Theorem 1.4 holds for what would be the case t = 4 in Theorem 1.3, which is not a Fermat pseudoprime base 2. Moreover, computations suggest that Theorem 1.3 holds for all t, not just for primes and Fermat pseudoprimes. Furthermore, it seems likely that a version of Theorem 1.4 exists for any family of bases congruent to 1 mod (2t − 1)2 for some t. 1.2 Lehmer's digit reversal problem D. H. Lehmer [8] considered sequences obtained by a procedure similar to Conway's RATS game. We begin by choosing an integer n and adding to it the number obtained by reversing the digits of n (no sorting or discarding of zeros). Lehmer asked, if we generate a sequence by iterating the above procedure, are we always guaranteed to eventually obtain a palindromic number? Lehmer's own work in base 10 shows that the answer is in the affirmative for positive integers smaller than 196. Large scale computations [10] strongly suggest that the answer is in the negative for most integers. My plan is to move beyond computational results, and to begin attacking this problem using a probabilistic model to explain the behavior of sequences generated using this procedure. 2 Other work 2.1 Rearrangements of number-theoretic series Over the last year I have assisted Bruce Berndt in providing proofs for a Ramanujan theta function transformation [1], as well as justifying previously unproven infinite series identities in Ramanu- jan's lost notebook (proofs to appear in the currently unpublished Ramanujan's Lost Notebook IV by Andrews and Berndt). In each case the proof has come down to showing that particular re- arrangements of certain number-theoretic series can be rigorously justified. My work in this area is currently still ongoing. I hope to apply my techniques to justify more of the remaining claims made in Ramanujan's lost notebook. 2.2 Arrangements of stars on a U.S. flag In an article in Slate, Wilson [11] introduces a notion of arrangements of stars on a U.S. flag that are \nice" in the sense of having certain symmetry properties. He showed that, with 3 exceptions, for all integers n ≤ 100 there exists a nice arrangement with n stars. Below are examples of the six different kinds of \nice" arrangements. In joint work with Dimitris Koukoulopoulos, we observed that the \niceness" property is closely related to the distribution of numbers in a multiplication table, a problem studied by Paul Erd¨os [3], Hall and Tenenbaum [7], and Ford [4]. We observed that a recent deep result of Ford on the distribution of divisors can be applied to the problem of existence of nice arrangements. We showed that, in contrast to what Wilson's empirical data might suggest, for almost all integers n, there exists no nice arrangement of n stars on the U.S. flag. I believe that this problem merits further computational investigation. Is there a way to measure how close a number is to having a nice star arrangement, even if it does not have one? What is the order of magnitude of the counting function of integers n ≤ x that have a nice star arrangement? These questions would make an excellent project for a student interested in number theory or computer science. References [1] Bruce C. Berdnt, Chadwick Gugg, Sarachai Kongsiriwong, and Johann Thiel. A proof of the general theta transformation formula. In Ramanujan Rediscovered: Proceedings of a Con- ference on Elliptic Functions, Partitions, and q-Series in memory of K. Venkatachaliengar: Bangalore, June 2009. [2] John H. Conway. Play it again..and again... Quantum, 63:30{31, November/December 1990. [3] P. Erdeˇs.` An asymptotic inequality in the theory of numbers. Vestnik Leningrad. Univ., 15(13):41{49, 1960. [4] Kevin Ford. The distribution of integers with a divisor in a given interval. Ann. of Math. (2), 168(2):367{433, 2008. [5] Karen Gentges. On Conway's RATS. Master's thesis, Central Missouri State University, 1998. [6] Richard K. Guy. Conway's RATS and other reversals. The American Mathematical Monthly, 96:425{428, 1989. [7] Richard R. Hall and G´eraldTenenbaum. Divisors, volume 90 of Cambridge Tracts in Mathe- matics. Cambridge University Press, Cambridge, 1988. [8] D. H. Lehmer. Sujets d'´etude. Sphinx (Bruxelles), 8:12{13. [9] Steven Shattuck and Curtis Cooper. Divergent RATS sequences. Fibonacci Quart., 39(2):101{ 106, 2001. [10] Wade VanLandingham. http://www.p196.org, 2006. [11] Chris Wilson. 13 stripes and 51 stars.