Gap estimates for the spectrum of $m$-bonacci numbers

Abstract

We establish some results on the structure of the spectrum of $m$-bonacci numbers. More precisely, we study explicit lower bounds for the distance between elements separated by $N$ positions in the ordered spectrum. The result is further detailed in the Fibonacci and Tribonacci case. The methodology combines the combinatorial structure of $m$-bonacci words and the canonical $m$-bonacci number system.