Title: Searching for Trott's constants

Abstract: It's well known that each real number may be represented as a continued fraction, as well as a decimal expansion. We investigate whether there is any intersection between these two representations. That is, are there any numbers whose continued fraction terms are the same as its decimal terms? The first to ask this question was M. Trott in 2006, a programmer at Mathematica, who found an interesting example

[0;1,0,8,4,1,0,1,5,1,2,2,...]=0.10841015122....

(where the left-hand side is in continued fraction notation). This number has an entry (here) in the Online Encyclopedia of Integer Sequences (oeis), it is usually called Trott's constant, and was found using a brute-force search algorithm. What's interesting about this number is that the terms in its decimal expansion and continued fraction expansion agree up to 639 terms! However, until recently, not much research has been done on this problem.

To give a rigorous phrasing to our question, we will formalize some definitions by saying precisely what it means for a number to have its decimal expansion agree with its continued fraction expansion. We call such numbers Trott constants, and give a method for their construction. We also give an efficient algorithm for computer searches. Through computer experiments, it appears that the set of Trott constants is a fractal, although we have yet to find proof of whether this set has positive dimension or not, or if it is even uncountable.