Restricted Partial Quotients - Restricted CFs and The Cantor Set

Restricted CFs and The Cantor Set

The Cantor set is a set C of measure zero from which a complete interval of real numbers can be constructed by simple addition – that is, any real number from the interval can be expressed as the sum of exactly two elements of the set C. The usual proof of the existence of the Cantor set is based on the idea of punching a "hole" in the middle of an interval, then punching holes in the remaining sub-intervals, and repeating this process ad infinitum.

The process of adding one more partial quotient to a finite continued fraction is in many ways analogous to this process of "punching a hole" in an interval of real numbers. The size of the "hole" is inversely proportional to the next partial denominator chosen – if the next partial denominator is 1, the gap between successive convergents is maximized.

To make the following theorems precise we will consider CF(M), the set of restricted continued fractions whose values lie in the open interval (0, 1) and whose partial denominators are bounded by a positive integer M – that is,

\mathrm{CF}(M) = \{: 1 \leq a_i \leq M \}.\,

By making an argument parallel to the one used to construct the Cantor set two interesting results can be obtained.

  • If M ≥ 4, then any real number in an interval can be constructed as the sum of two elements from CF(M), where the interval is given by
(2\times, 2\times) = \left(\frac{1}{M} \left, \sqrt{M^2 + 4M} - M \right).
  • A simple argument shows that holds when M ≥ 4, and this in turn implies that if M ≥ 4, every real number can be represented in the form n + CF1 + CF2, where n is an integer, and CF1 and CF2 are elements of CF(M).

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