A quantitative version of the Absolute Subspace Theorem

On two notions of complexity of algebraic numbers

“…Now conditions (A.1)-(A.7) imply the conditions (5.12)-(5.17) on p. 36 of [14] with n = 2. These conditions are kept throughout [14] and so all arguments of [14] from p. 36 onwards are applicable in our situation. Since in what follows the tuples L and c will be fixed and only Q will vary, we will write H Q for the twisted height H Q,L,c .…”

“…For n ≥ 3 this is precisely Theorem 2.1 of [14], while for n = 2 this is an improvement of that theorem. This improvement can be obtained by combining some lemmata from [14] with more precise computations in the case n = 2.…”

“…This improvement can be obtained by combining some lemmata from [14] with more precise computations in the case n = 2. We give more details in the appendix at the end of the present paper.…”

“…In [14] (specialized to n = 2), λ 1 (Q), λ 2 (Q) were defined to be the successive infima of some sort of parallelepiped Π(Q, c) defined over Q, and the lemmata in that paper were all formulated in terms of those infima. However, according to [14,Corollary 7.4,p.…”

“…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 version of the quantitative subspace theorem by Evertse and Schlickewei [14,Theorem 2.1].…”