Combinatorical and Algebraic Structures Seminar

Session details

Date: 30.10.2012
Speaker: Tomáš Vávra, FJFI, České vysoké učení technické v Praze
Title: On confluence and Erdos-Komornik-Joo problem in negative base
Abstract: In positive base number systems many properties are specific for the class confluent Pisot bases, i.e. zeros of $x^k-mx^{k-1}-mx^{k-2}-\cdots-mx-n$, where $k\geq 1$, $m\geq n\geq 1$. The main aspect –- giving them their name –- is that any integer combinations of non-negative powers of the base with coefficients in $\{0,1,\dots,\lceil\beta\rceil-1\}$ is a $\beta$-integer, although the sequence of coefficients may be forbidden in the corresponding number system. Among other exceptional aspects is the existence of optimal representations, or the fact that the infinite word $u_\beta$ coding the $\beta$-integers is reversal closed and the corresponding Rauzy fractal is centrally symmetric. Confluent Pisot bases are also among the only cases where the Erdos-Komornik-Joo problem has been solved. We concentrate on the question of confluence in negative base systems. We show that any integer combinations of non-negative powers of the base with coefficients in $\{0,1,\dots,\lfloor\beta\rfloor\}$ is a $(-\beta)$-integer if and only if $\beta$ is a zero of the above polynomial satisfying $m=n$ when $k$ is even. It turns out that for such bases, the infinite word $u_{-\beta}$ coding $(-\beta)$-integers has the same language as that of $u_\beta$ and consequently, the corresponding Rauzy fractals coincide up to translation. It is likely that these are the only bases where such a coincidence between positive and negative base systems happens. As a consequence of our result, one can solve an instance of the Erdos-Komornik-Joo problem generalized to negative bases.

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