Bounded Sequence Of Integers And Transcendence Of Real Numbers

The problem of determining the transcendence of real numbers is animportant issue in the number theory. It is well known that almost all realnumbers are transcendental numbers, and no e?ective transcendence criteriaexist. The representation of real numbers, either its continued fraction ex-pansion or b-adic expansion, is a sequence of nonnegative integer numbers.Moreover, the sequence in the latter case is bounded. In this paper we dis-cuss the transcendence of the continued fractions with bounded quotients aswell as the b-adic numbers by the construction of bounded integer sequences.In this paper, we ?rst outline a number of prior knowledge, includingsome basic concepts of words and sequences, the concepts and basic proper-ties of continued fractions, and the relative knowledge of normal numbers.Then we present some results about the transcendence of continued fractionsand b-adic numbers, and apply these results to some special sequences. No-tice that the corresponding continued fractions and b-adic numbers are alltranscendental when the sequences are Sturmian sequences or Thue-Morsesequences. In this paper we focus our attention on the sequences to whichcorresponding continued fractions and b-adic numbers are both transcen-dental. In the end, we raise some questions.We test and verify some results in this paper: 1, For any irrationalθand integer d(≥2), if the expansion in base b(≥d) of a positive real numberαis ( [nθ]modd )n≥1 , thenαis transcendental. 2, For any positive integerb≥2 and non-negative integers a < c < b , if the expansion in base b of apositive real numberαis a Rudin-Shapiro sequence or Cantor sequence onthe alphabet {a,c}, thenαis transcendental. 3, For any positive integera < b , if (an)n≥0 is a Cantor sequence on the alphabet {a,b}, then thecontinued fraction [a0; a1,a2,...] is transcendental.