HAUSDORFF DIMENSION ANALYSIS OF SETS WITH THE PRODUCT OF CONSECUTIVE VS SINGLE PARTIAL QUOTIENTS IN CONTINUED FRACTIONS

We present a detailed Hausdorff dimension analysis of the set of real numbers where the product of consecutive partial quotients in their continued fraction expansion grows at a certain rate but the growth of the single partial quotient is at a different rate. We consider the set 7.(f1, f2) def = 6(f1)\6(f2) = ( ) an(x)an & PLUSMN;1(x) > f1(n) for infinitely many n E N x E [0, 1) : , an & PLUSMN;1(x) < f2(n) for all sufficiently large n E N where fi : N-+ (0, oo) are any functions such that lim n & RARR;& INFIN; some surprising results including the situations when 7.(f1, f2) is empty for various non-trivial choices of fi's. Our results contribute to the metrical theory of continued fractions by generalising several known results including the main result of [Nonlinearity, 33(6):2615-2639, 2020]. To obtain some of the results, we consider an alternate generalised set, which may be of independent interest, and calculate its Hausdorff dimension. One of the main ingredients is the introduction of an idea of two different types of probability measures supported on a suitably constructed Cantor type subset and then using the classical mass distribution principle. fi(n) = oo. We obtain