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Scales of Functions and Applications in Ergodic Theory and Number Theory.

4:00 pm Thursday, September 10, 2015
Michael Boshernitzan (Rice)

G. H. Hardy in his book ”Orders of Infinity” introduced a class L of real functions defined in a neighborhood of +∞ by means of certain formula involving the variable x, real constants, algebraic operations and the functional symbols exp() and log(). He proved that this class forms a scale: it is linearly ordered by eventual dominance at +∞.

We describe how some abstract definitions lead unexpectedly to various classes of scales, some much larger than L. The functions in these scales satisfy various differential, difference and functional equations. We present some known results on the structure of these classes and state a number of open questions and conjectures.

Discrete versions of the above construction leads to large families of ”good” sequences (e.g. sequences generated by certain recurrence relations) comparable with each other.

Some applications of these results to the questions on uniform distribution in Ergodic Theory and Number Theory will be described, as well as the connection to the 16-th Hilbert problem.

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