Experimental Mathematics

Approximate Squaring

J. C. Lagarias and N. J. A. Sloane

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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.

Article information

Source
Experiment. Math., Volume 13, Number 1 (2004), 113-128.

Dates
First available in Project Euclid: 10 June 2004

Permanent link to this document
https://projecteuclid.org/euclid.em/1086894093

Mathematical Reviews number (MathSciNet)
MR2065571

Zentralblatt MATH identifier
1115.11045

Subjects
Primary: 26A18: Iteration [See also 37Bxx, 37Cxx, 37Exx, 39B12, 47H10, 54H25]
Secondary: 11B83: Special sequences and polynomials 11K31: Special sequences 11Y99: None of the above, but in this section

Keywords
Approximate squaring approximate multiplication iterated maps integer sequences 3x + 1 problem Mahler z-numbers

Citation

Lagarias, J. C.; Sloane, N. J. A. Approximate Squaring. Experiment. Math. 13 (2004), no. 1, 113--128. https://projecteuclid.org/euclid.em/1086894093


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