All a b c d e f g h i j k l m n o p q r s t u v w x y z Other

Previous Page 38 Next

Displaying 741 – 760 of 1964

Showing per page

A problem of Galambos on Engel expansions

Jun Wu (2000)

Acta Arithmetica

1. Introduction. Given x in (0,1], let x = [d₁(x),d₂(x),...] denote the Engel expansion of x, that is, (1) x = 1 / d ₁ ( x ) + 1 / ( d ₁ ( x ) d ₂ ( x ) ) + . . . + 1 / ( d ₁ ( x ) d ₂ ( x ) . . . d n ( x ) ) + . . . , where d j ( x ) , j ≥ 1 is a sequence of positive integers satisfying d₁(x) ≥ 2 and d j + 1 ( x ) ≥ d j ( x ) for j ≥ 1. (See [3].) In [3], János Galambos proved that for almost all x ∈ (0,1], (2) l i m n → ∞ d n 1 / n ( x ) = e . He conjectured ([3], P132) that the Hausdorff dimension of the set where (2) fails is one. In this paper, we prove this conjecture: Theorem. d i m H x ∈ ( 0 , 1 ] : ( 2 ) f a i l s = 1 . We use L¹ to denote the one-dimensional Lebesgue measure on (0,1] and d i m H to denote the Hausdorff...

A problem of Rankin on sets without geometric progressions

Melvyn B. Nathanson, Kevin O'Bryant (2015)

Acta Arithmetica

A geometric progression of length k and integer ratio is a set of numbers of the form a , a r , . . . , a r k - 1 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 ( a i ) i = 1 ∞ of positive real numbers with a₁ = 1 such that the set G ( k ) = ⋃ i = 1 ∞ ( a 2 i , a 2 i - 1 ] contains no geometric progression of length k and integer ratio. Moreover, G ( k ) 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...

Currently displaying 741 – 760 of 1964