Essential difference between the repetitive thereshold and asymptotic repetitive threshold of balanced sequences

Lubomíra Dvořáková
Czech Technical University in Prague
https://kmlinux.fjfi.cvut.cz/~balkolub/

Date(s) : 23/01/2023 iCal
15 h 00 min - 16 h 00 min

At first, we will summarize both the history and the state of the art of the critical exponent and the asymptotic critical exponent of balanced sequences. Second, we will colour the Fibonacci sequence by suitable constant gap sequences to provide an upper bound on the asymptotic repetitive threshold of $d$ -ary balanced sequences. The bound is attained for $d$ equal to $2$ , $4$ and $8$ and we conjecture that it happens for infinitely many even $d$ ‘s. Finally, we will reveal an essential difference in behavior of the repetitive threshold and the asymptotic repetitive threshold of balanced sequences: The repetitive threshold of $d$ -ary balanced sequences is known to be at least $1 + 1 / (d - 2)$ for each $d$ larger than two. In contrast, our bound implies that the asymptotic repetitive threshold of $d$ -ary balanced sequences is at most $1 + ϕ^{3} / 2^{d - 3}$ for each $d$ larger than one, where $ϕ$ is the golden mean.

Joint work with Edita Pelantová.

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Virtual event

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