Any upper bound for $a_i$ in $\gamma =\{a_0,a_1,\dots,a_i,\dots\}$ the simple continued fraction expansion of real positive algebraic numbers?

Question

Are there any upper bound for $a_i$ in $\gamma =\{a_0,a_1,\dots,a_i,\dots\}$ the simple continued fraction expansion of real positive algebraic numbers?

Answer 1

Does not seem likely. I am looking at Khinchin's little book. Quadratic irrationals have bounded ""elements" because they are eventually periodic. Meanwhile, Theorem 23 on page 36, bounded elements implies only finitely many convergents with error less than $c/q^2.$ Put that with Thue-Siegel-Roth, i think higher degree algebraic have unbounded elements.

Hmmm. I may have that backwards, TSR may add nothing in this direction. Things to do, if I come up with anything else...