Approximate Squaring

Abstract

We study the "approximate squaring'' map {\small $f(x) := x \lceil x \rceil$} and its behavior when iterated. We conjecture that if f is repeatedly applied to a rational number {\small $r = l/d > 1$} then eventually an integer will be reached. We prove this when {\small $d=2$}, 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 {\small $f_r(x) := r \lceil x \rceil$}, where r is a fixed rational number. We briefly discuss what happens when "ceiling'' is replaced by "floor'' in the definitions.