Prescribed realisation of longest runs in continued fractions

Abstract

Exceptional longest-run behaviour in continued fraction expansions is studied through the interaction between fixed-symbol runs and the overall longest run. For every prescribed partial quotient value and every admissible growth scale, a full Hausdorff dimensional set of irrational numbers is constructed on which the longest run of the prescribed value has exactly the prescribed asymptotic growth and, for every initial length, uniquely realises the overall maximum. It follows that the symbol responsible for the overall longest run can be fixed in advance without any loss of Hausdorff dimension. Thus, the known full-dimensional exceptional-set results for fixed-symbol longest-run growth and for overall longest-run growth are simultaneously strengthened, while the maximising symbol in the overall problem is shown to be fully prescribable.