Sums of nearest integer continued fractions with bounded digits: $\textrm{NICF}_5 + \textrm{NICF}_5 = {\mathbb R}$

TL;DR

The paper proves that every real number can be expressed as a sum of two NICF-bounded numbers with partial quotients bounded by $5$, extending Cantor-set methods to nearest-integer continued fractions. It constructs a NICF-based Cantor set $C_{\operatorname{NICF}}$ with density ratio exceeding 1 and shows $C_{\operatorname{NICF}}+C_{\operatorname{NICF}}$ contains $[\tfrac{1}{2},\tfrac{3}{2}]$, yielding $\mathbb{R}=\operatorname{NICF}_5+\operatorname{NICF}_5$. The result is sharp: replacing 5 with 4 fails, as demonstrated by a detailed analysis of NICF$_4$-related sets, their sums, and measure properties. The work highlights the power of Cantor-set gap-analysis and the notions of comparable/dividable intervals to control sumsets of bounded-NICF expansions, and it connects to classical Hall-Hlawka/Hlavka results for regular continued fractions. These methods potentially inform extensions to complex or Hurwitz-type continued fractions.

Abstract

Adapting Cantor set methods that were used by Hall and Hlawka for regular continued fractions, we prove that every real number can be obtained as the sum of two real numbers for which the partial fractions in their nearest integer continued fraction expansion do not exceed 5 (with the zeroth partial fraction as the only possible exception). Furthermore, we prove that it is not possible to replace 5 by 4 in this result.

Figures (6)

• Figure 1: Gap for $T_{2, 5}[{\bar{a}}]$, with ${\bar{a}} = [a_0;a_1, \ldots, a_n ]$, for $n$ odd and $n$ even
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Theorems & Definitions (47)

• Theorem 1: Hall Hall, 1947
• Theorem 2
• Lemma 3
• Definition 4
• Definition 5
• Proposition 6
• Lemma 7
• proof
• Theorem 8
• Definition 9