A Discrete Fourier Kernel and Fraenkel's Tiling Conjecture

The set B_{p,r}^q:=\{\floor{nq/p+r} \colon n\in Z \} with integers p, q, r) is a Beatty set with density p/q. We derive a formula for the Fourier transform \hat{B_{p,r}^q}(j):=\sum_{n=1}^p e^{-2 \pi i j \floor{nq/p+r} / q}. A. S. Fraenkel conjectured that there is essentially one way to partition the integers into m>2 Beatty sets with distinct densities. We conjecture a generalization of this, and use Fourier methods to prove several special cases of our generalized conjecture.Comment: 24 pages, 6 figures (now with minor revisions and clarifications

A Model for Pairs of Beatty Sequences

Two Beatty sequences are recorded by athletes running in opposite directions in a round stadium. This approach suggests a nice interpretation for well known partitioning criteria: such sequences (eventually) partition the integers essentially when the athletes have the same starting point

Self-Matching Properties of Beatty Sequences

We study the selfmatching properties of Beatty sequences, in particular of the graph of the function $\lfloor j\beta\rfloor$ against $j$ for every quadratic unit $\beta\in(0,1)$. We show that translation in the argument by an element $G_i$ of generalized Fibonacci sequence causes almost always the translation of the value of function by $G_{i-1}$. More precisely, for fixed $i\in\N$, we have $\bigl\lfloor \beta(j+G_i)\bigr\rfloor = \lfloor \beta j\rfloor +G_{i-1}$, where $j\notin U_i$. We determine the set $U_i$ of mismatches and show that it has a low frequency, namely $\beta^i$.Comment: 7 page

A characterization of covering equivalence

Let A={a_s(mod n_s)}_{s=1}^k and B={b_t(mod m_t)}_{t=1}^l be two systems of residue classes. If |{1\le s\le k: x=a_s (mod n_s)}| and |{1\le t\le l: x=b_t (mod m_t)}| are equal for all integers x, then A and B are said to be covering equivalent. In this paper we characterize the covering equivalence in a simple and new way. Using the characterization we partially confirm a conjecture of R. L. Graham and K. O'Bryant

A characterization of balanced episturmian sequences

It is well known that Sturmian sequences are the aperiodic sequences that are balanced over a 2-letter alphabet. They are also characterized by their complexity: they have exactly $(n+1)$ factors of length $n$. One possible generalization of Sturmian sequences is the set of infinite sequences over a $k$-letter alphabet, $k \geq 3$, which are closed under reversal and have at most one right special factor for each length. This is the set of episturmian sequences. These are not necessarily balanced over a $k$-letter alphabet, nor are they necessarily aperiodic. In this paper, we characterize balanced episturmian sequences, periodic or not, and prove Fraenkel's conjecture for the class of episturmian sequences. This conjecture was first introduced in number theory and has remained unsolved for more than 30 years. It states that for a fixed $k> 2$, there is only one way to cover $\Z$ by $k$ Beatty sequences. The problem can be translated to combinatorics on words: for a $k$-letter alphabet, there exists only one balanced sequence up to letter permutation that has different letter frequencies

Wythoff Wisdom

International audienceSix authors tell their stories from their encounters with the famous combinatorial game Wythoff Nim and its sequences, including a short survey on exactly covering systems