Transcendence criteria for pairs of continued fractions

GLASNIK MATEMATI?KI Vol. 41(61)(2006), 223 231 TRANSCENDENCE CRITERIA FOR PAIRS OF CONTINUED FRACTIONS Boris Adam zewski and Yann Bugeaud Université Claude Bernard Lyon 1 and Université Louis Pasteur, Fran e Abstra t. The aim of the present note is to establish two extensions of some trans enden e riteria for real numbers given by their ontinued fra tion expansions. We adopt the following point of view: rather than giving su ient onditions ensuring the trans enden e of a given number ?, we take a pair (?, ??) of real numbers, and we prove that, under some ondition, at least one of them is trans endental. 1. Introdu tion and results Very little is known on the ontinued fra tion expansion of any algebrai real number of degree at least three. It is likely that the sequen e of its partial quotients is unbounded, but we seem to be still very far away from a proof. Re ently, a small step was made in this dire tion by means of several new trans enden e riteria for ontinued fra tions [1, 2, 3?. They illustrate the fa t that if the sequen e of partial quotients of a real irrational number ? has some spe ial ombinatorial property, for example if long blo ks of partial quotients repeat unusually lose to the beginning, then ? must be either trans endental, or quadrati .

• let


• nite words


• qsn?1 qsn


• trans enden


• dire tion


• pn?1


• either trans


• positive integers