TY - JOUR

T1 - Generalizations of Sturmian sequences associated with N-continued fraction algorithms

AU - Langeveld, Niels

AU - Rossi, Lucia

AU - Thuswaldner, Jörg

PY - 2023/9

Y1 - 2023/9

N2 - Given a positive integer N and x ∈ [0, 1] \ Q, an N-continuedfraction expansion of x is defined analogously to the classicalcontinued fraction expansion, but with the numerators beingall equal to N. Inspired by Sturmian sequences, we introducethe N-continued fraction sequences ω(x, N) and ω(x, N),which are related to the N-continued fraction expansion ofx. They are infinite words over a two letter alphabet obtainedas the limit of a directive sequence of certain substitutions,hence they are S-adic sequences. When N = 1, we are in thecase of the classical continued fraction algorithm, and obtainthe well-known Sturmian sequences. We show that ω(x, N)and ω(x, N) are C-balanced for some explicit values of Cand compute their factor complexity function. We also obtainuniform word frequencies and deduce unique ergodicity of theassociated subshifts. Finally, we provide a Farey-like map forN-continued fraction expansions, which provides an additiveversion of N-continued fractions, for which we prove ergodicityand give the invariant measure explicitly

AB - Given a positive integer N and x ∈ [0, 1] \ Q, an N-continuedfraction expansion of x is defined analogously to the classicalcontinued fraction expansion, but with the numerators beingall equal to N. Inspired by Sturmian sequences, we introducethe N-continued fraction sequences ω(x, N) and ω(x, N),which are related to the N-continued fraction expansion ofx. They are infinite words over a two letter alphabet obtainedas the limit of a directive sequence of certain substitutions,hence they are S-adic sequences. When N = 1, we are in thecase of the classical continued fraction algorithm, and obtainthe well-known Sturmian sequences. We show that ω(x, N)and ω(x, N) are C-balanced for some explicit values of Cand compute their factor complexity function. We also obtainuniform word frequencies and deduce unique ergodicity of theassociated subshifts. Finally, we provide a Farey-like map forN-continued fraction expansions, which provides an additiveversion of N-continued fractions, for which we prove ergodicityand give the invariant measure explicitly

U2 - 10.1016/j.jnt.2023.03.008

DO - 10.1016/j.jnt.2023.03.008

M3 - Article

VL - 250.2023

SP - 49

EP - 83

JO - Journal of number theory

JF - Journal of number theory

SN - 0022-314X

IS - September

ER -