A problem involving simultaneous binary compositions
1. Introduction. Given x in (0,1], let x = [d₁(x),d₂(x),...] denote the Engel expansion of x, that is, (1) , where is a sequence of positive integers satisfying d₁(x) ≥ 2 and for j ≥ 1. (See [3].) In [3], János Galambos proved that for almost all x ∈ (0,1], (2) He conjectured ([3], P132) that the Hausdorff dimension of the set where (2) fails is one. In this paper, we prove this conjecture: Theorem. . We use L¹ to denote the one-dimensional Lebesgue measure on (0,1] and to denote the Hausdorff...
A geometric progression of length k and integer ratio is a set of numbers of the form for some positive real number a and integer r ≥ 2. For each integer k ≥ 3, a greedy algorithm is used to construct a strictly decreasing sequence of positive real numbers with a₁ = 1 such that the set contains no geometric progression of length k and integer ratio. Moreover, is a maximal subset of (0,1] that contains no geometric progression of length k and integer ratio. It is also proved that there is...