It appears Niederreiter's theorem can be improved. Consider the following sequence of rationals (numerators in hexadecimal) 1/2^1 5/2^3 4F/2^7 4F01/2^15 4F00FFFF/2^31 4F00FFFF00000001/2^63 4F00FFFF00000000FFFFFFFFFFFFFFFF/2^127 ... in each case, the denominator is 2^(2^k-1) and the numerator is got by either (i) subtracting 1 from previous numerator then appending 1111...1111 in binary, or (i) appending 0000...0001 to the previous numerator's binary form (the append always exactly doubles bitlength). where we do (i) if k=odd and (ii) if k=even. In other words, the numerator is 2^(2^(k-1)) times the preceding numerator, plus (-1)^k. All these rationals have only 1s and 2s in their continued fraction expansion. No 3s. There are many other example sequences of the same ilk, where no 3's occur. This one just happens to be the simplest. Their continued fraction expansions are [0; 2] = [0; 1, 1] (Note, any final "2" can be replaced by "1, 1") [0; 1, 1, 1, 2] [0; 1, 1, 1, 1, 1, 1, 2, 1, 2] [0; 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2] [0; 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2] [0; 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2] [0; 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2] the rule for going from one CF expansion to the next is either you (i) copy its after-semicolon part (in form ending 2) twice, or (ii) copy it once (in form ending 1,1) then subtract 1 from the final entry; then append "1", then append a REVERSE-ORDER copy (of the form ending 2). In other words, Niederreiter's construction actually is stronger than he realized, since it creates infinite chains of rationals with denominator=power of 2, having only 1s and 2s in their continued fraction expansions. This is amazing. But then I found this paper: https://web.williams.edu/Mathematics/sjmiller/public_html/book/papers/vdp/Po... FOLDED CONTINUED FRACTIONS by A. J. van der Poorten and J. Shallit which seems to know about this sort of thing already. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)