Products and quotients of numbers with small partial quotients

Stephen Astels

Displaying similar documents to “Products and quotients of numbers with small partial quotients”

Uniform distribution modulo one and binary search trees

Michel Dekking, Peter Van der Wal (2002)

Journal de théorie des nombres de Bordeaux

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Any sequence $x = {(x_{k})}_{k = 1}^{\infty}$ of distinct numbers from [0,1] generates a binary tree by storing the numbers consecutively at the nodes according to a left-right algorithm (or equivalently by sorting the numbers according to the Quicksort algorithm). Let $H_{n} (x)$ be the height of the tree generated by $x_{1}, \dots, x_{n} .$ Obviously $\frac{log n}{log 2} - 1 \leq H_{n} (x) \leq n - 1 .$ If the sequences $x$ are generated by independent random variables having the uniform distribution on [0, 1], then it is well known that there exists $c$ > $0$ such that...

New bounds on the length of finite pierce and Engel series

P. Erdös, J. O. Shallit (1991)

Journal de théorie des nombres de Bordeaux

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Every real number $x, 0 < x \leq 1$ , has an essentially unique expansion as a Pierce series : $x = \frac{1}{x^{1}} - \frac{1}{x^{1} x^{2}} + \frac{1}{x^{1} x^{2} x^{3}} - \dots$ where the $x_{i}$ form a strictly increasing sequence of positive integers. The expansion terminates if and only if $x$ is rational. Similarly, every positive real number $y$ has a unique expansion as an Engel series : $y = \frac{1}{y^{1}} - \frac{1}{y^{1} y^{2}} + \frac{1}{y^{1} y^{2} y^{3}} + \dots$ where the $y_{i}$ form a (not necessarily strictly) increasing sequence of positive integers. If the expansion is infinite, we require that the sequence yi...

On the binary expansions of algebraic numbers

David H. Bailey, Jonathan M. Borwein, Richard E. Crandall, Carl Pomerance (2004)

Journal de Théorie des Nombres de Bordeaux

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Employing concepts from additive number theory, together with results on binary evaluations and partial series, we establish bounds on the density of 1’s in the binary expansions of real algebraic numbers. A central result is that if a real $y$ has algebraic degree $D > 1$ , then the number $# (| y |, N)$ of 1-bits in the expansion of $| y |$ through bit position $N$ satisfies $# (| y |, N) > C N^{1 / D}$ for a positive number $C$ (depending on $y$ ) and sufficiently large $N$ . This in itself establishes the transcendency...

The bilinear Hilbert trasform is pointwise finite.

Michael T. Lacey (1997)

Revista Matemática Iberoamericana

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Let f ∈ L^∞ and g ∈ L² be supported on [0,1]. Then the principal value integral below exists in L¹. p.v. ∫ f(x + y) g(x - y) dy / y.

A remark on branch weights in countable trees

Bohdan Zelinka (2004)

Mathematica Bohemica

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Let $T$ be a tree, let $u$ be its vertex. The branch weight $b (u)$ of $u$ is the maximum number of vertices of a branch of $T$ at $u$ . The set of vertices $u$ of $T$ in which $b (u)$ attains its minimum is the branch weight centroid $B (T)$ of $T$ . For finite trees the present author proved that $B (T)$ coincides with the median of $T$ , therefore it consists of one vertex or of two adjacent vertices. In this paper we show that for infinite countable trees the situation is quite different.

Some new classes of graceful Lobsters obtained from diameter four trees

Debdas Mishra, Pratima Panigrahi (2010)

Mathematica Bohemica

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We observe that a lobster with diameter at least five has a unique path $H = x_{0}, x_{1}, ..., x_{m}$ with the property that besides the adjacencies in $H$ both $x_{0}$ and $x_{m}$ are adjacent to the centers of at least one $K_{1, s}$ , where $s > 0$ , and each $x_{i}$ , $1 \leq i \leq m - 1$ , is adjacent at most to the centers of some $K_{1, s}$ , where $s \geq 0$ . This path $H$ is called the central path of the lobster. We call $K_{1, s}$ an even branch if $s$ is nonzero even, an odd branch if $s$ is odd and a pendant branch if $s = 0$ . In the existing literature only some specific classes of lobsters have...