Hofstadter’s Conjecture for $\alpha = \sqrt 2 - 1$

Abstract

The goal of this paper is to generalize the concepts and methods used in the study of Hofstadter’s extraction conjecture begun in [2] and [5]. Let α, 0 < α < 1, be irrational, let x = x(α) be the infinite string whose n-th element is “c” or “d” depending on whether [(n + l)α] − [na] equals 0 or 1 respectively, with [z] denoting the greatest integer function. For integer m ≥ 0 define s _m , x _m by

$$ x = {s_m}{x_m}, L\left( {{s_m}} \right) = m $$

((1))

with L(s) denoting the length of the string s.