<abstract xmlns="http://www.w3.org/1999/xhtml"> This expository paper presents known results on distribution functions g(x) of the sequence of blocks <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/Untitled-1.jpg"></inline-graphic></alternatives></inline-formula> where xn is an increasing sequence of positive integers. Also presents results of the set G(Xn) of all distribution functions g(x). Specially:- continuity of g(x);- connectivity of G(Xn);- singleton of G(Xn);- one-step g(x);- uniform distribution of Xn, n = 1, 2, . . . ;- lower and upper bounds of g(x);- applications to bounds of <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/Untitled-2.jpg"></inline-graphic></alternatives></inline-formula>;- many examples, e.g., <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/Untitled-3.jpg"></inline-graphic></alternatives></inline-formula>, where pn is the nth prime, is uniformly distributed.The present results have been published by 25 papers of several authors between 2001-2013. </abstract>

This expository paper presents known results on distribution functions g(x) of the sequence of blocks  where xn is an increasing sequence of positive integers. Also presents results of the set G(Xn) of all distribution functions g(x). Specially:- continuity of g(x);- connectivity of G(Xn);- singleton of G(Xn);- one-step g(x);- uniform distribution of Xn, n = 1, 2, . . . ;- lower and upper bounds of g(x);- applications to bounds of ;- many examples, e.g., , where pn is the nth prime, is uniformly distributed.The present results have been published by 25 papers of several authors between 2001-2013.

Distribution functions of ratio sequences. An expository paper

Mathematical Institute Slovak Academy of Sciences Štefánikova 49 SK–814-73 Bratislava SLOVAKIA

Tatra Mountains Mathematical Publications

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

{"article-title":"Distribution functions of ratio sequences. An expository paper"}

This expository paper presents known results on distribution functions g(x) of the sequence of blocks  where xn is an increasing sequence of positive integers....

Distribution functions of ratio sequences. An expository paper

Published Online: Feb 19, 2016

Page range: 133 - 185

Received: Nov 28, 2015

DOI: https://doi.org/10.1515/tmmp-2015-0047

Keywords
block sequence, distribution function, asymptotic density

© 2016

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Distribution functions of ratio sequences. An expository paper

Published Online: Feb 19, 2016

Page range: 133 - 185

Received: Nov 28, 2015

DOI: https://doi.org/10.1515/tmmp-2015-0047

Keywordsblock sequence, distribution function, asymptotic density

© 2016

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Keywords
block sequence, distribution function, asymptotic density