On two notions of complexity of algebraic numbers

Abstract

we derive new, improved lower bounds for the block complexity of an irrational algebraic number and for the number of digit changes in the b-ary expansion of an irrational algebraic number. To this end, we apply a quantitative version of the Subspace Theorem due to Evertse and Schlickewei (2002).Comment: 31 page

Automatic continued fractions are transcendental or quadratic

“…Furthermore, proceeding as in [15] and in [18], it seems to be possible to prove that if a = a 1 a 2 . .…”

An explicit lower bound for the block complexity of an algebraic number

“…Now, we discuss consequences of Lemma 1 and the Parametric Subspace Theorem [7]. We content ourself to sketch the proofs of our theorems, since they are very similar to that of Theorem 2.1 from [6].…”

“…Arguing as in [6], we establish that all the vectors x n , ⌊N/2⌋ ≤ n ≤ N , lie in the union of at most 10 160 c 8 log(8d) log log(8d)…”

“…. < i r } be the set of n with ⌊N/2⌋ ≤ n ≤ N such that x n is in H. Arguing again as in [6] and using (3.1), it follows that r ≤ 10 20 c 3 (log c) log(8d) log log(8d)…”

“…where we have set t n = r n + s n , satisfies a system of inequalities to which we can apply the Parametric Subspace Theorem with ε = 1/(5c), exactly as in [6]. We use an explicit estimate for the number of subspaces that contain all the solutions having a sufficiently large height.…”