Approximate Squaring

Details Approximate Squaring Approximate Squaring

2003-09-23

arxiv.org/abs/math/0309389v2

Experimental Math. 13 (2004), 113--128.

Mathematics

Number Theory

22 pages. Revised Nov 9, 2003: new theorems, including probabilistic interpretation of results, also analogs for floor functio

Scientific paper

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.

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