On the intersections of exceptional sets in Borel's normal number theorem and Erdoes-Renyi limit theorem

摘要

For any x is an element of [0, 1), let S-n(x) be the partial summation of the first n digits in the binary expansion of x and R-n(x) be its run-length function. The classical Borel's normal number theorem tells us that for almost all x is an element of [0, 1), the limit of S-n(x)/n as n goes to infinity is one half. On the other hand, the Erdos-Renyi limit theorem shows that R-n(x) increases to infinity with the logarithmic speed log(2) n as n - infinity for almost every x in [0, 1). In this paper, we are interested in the intersections of exceptional sets arising in the above two famous theorems. More precisely, for any 0 = alpha(1) = alpha(2) = 1 and 0 = beta(1) = beta(2) = +infinity, we completely determine the Hausdorff dimension of the following set:B(alpha(1), alpha(2)) boolean AND E(beta(1), beta(2)),whereB(alpha(1), alpha(2)) = {x is an element of [0, 1) : lim inf(n -infinity) S-n(x)/n = alpha(1), lim sup(n -infinity) S-n(x)/n = alpha(2)}andE(beta(1), beta(2)) = {x is an element of [0, 1) : lim inf(n -infinity) R-n(x)/log(2) n = beta(1), lim sup(n -infinity) R-n(x)/log(2) n = beta(2)}.After some minor modifications, our result still holds if we replace the denominator log(2) n in E(beta(1), beta(2)) with any increasing function phi : N - R+ satisfying phi(n) tending to +infinity and lim(n -infinity)(phi(n + 1) - phi(n)) = 0. As a result, we also obtain that the set of points for which neither the sequence {S-n(x)/n}(n = 1) nor {R-n(x)/phi(n)}(n = 1) converges has full Hausdorff dimension.