Continued fraction Cantor sets, Hausdorff dimension, and functional analysis

Continuity of the Hausdorff Measure of Continued Fractions and Countable Alphabet Iterated Function Systems

“…, is a finite conformal iterated function system consisting of at least two elements, we have (see [1] or [4] for instance) the following well-known result.…”

“…, n}. It is well-known (see [1], comp. [3] and [4], where an analogous statement is proved for all conformal iterated function systems), that n(1 − HD J n (G) ) = 6 π 2 .…”

“…Motivated by such results and some continuity properties of the numerical value of the Hausdorff measure of the limit sets in conformal dynamics (see [5] and [6]), we asked ourselves whether a continuity like in (1.1) holds on a deeper level of Hausdorff measures. Armed with the theory of iterated function systems it can be relatively easy to show that the continuity holds for normalized Hausdorff measures in the weak * topology on Borel probability measures on the unit interval [0,1]. For the numerical values of Hausdorff measures the positiver answer is given in Section 3 below; see Theorem 3.1.…”

О множествах вида $\mathscr A+\mathscr B$ и конечных цепных дробях

“…принадлежащее Д. Хенсли [9]. Отметим, что в более ранней работе Д. Хенс-ли [10] имеется более простое неравенство…”

“…Основными результатами настоящей работы являются асимптотические оценки для количества элементов в пересечении B − r (t 1 , p)∩B + r (t 2 , p) при условиях более слабых, чем (9). В частности, нам удалось получить асимптотики даже тогда, когда одно из чисел β j близко к 1/2.…”