Approximate Squaring
Number Theory
2007-07-16 v2 Information Theory
math.IT
Abstract
We study the ``approximate squaring'' map f(x) := x ceiling(x) and its behavior when iterated. We conjecture that if f is repeatedly applied to a rational number r = l/d > 1 then eventually an integer will be reached. We prove this when d=2, and provide evidence that it is true in general by giving an upper bound on the density of the ``exceptional set'' of numbers which fail to reach an integer. We give similar results for a p-adic analogue of f, when the exceptional set is nonempty, and for iterating the ``approximate multiplication'' map f_r(x) := r ceiling(x) where r is a fixed rational number.
Cite
@article{arxiv.math/0309389,
title = {Approximate Squaring},
author = {J. C. Lagarias and N. J. A. Sloane},
journal= {arXiv preprint arXiv:math/0309389},
year = {2007}
}
Comments
22 pages. Revised Nov 9, 2003: new theorems, including probabilistic interpretation of results, also analogs for floor function (24 pages)