Sums of numbers with small partial quotients (Q2759007)

For a positive integer \(m\) let \(F(m)\) be the set of numbers \(F(m)=\{[t,a_1,a_2,\dots]; t\in\mathbb Z, 1\leq a_i\leq m (i\geq 1)\}\), where \([a_0,a_1,a_2,\dots]\) denotes the continued fraction expansion. \textit{J. Hlavka} [Trans. Am. Math. Soc. 211, 123-134 (1975; Zbl 0313.10032)] stated that \(F(3)+F(2)+F(2)\neq\mathbb R\). In this paper the author shows that this is false by establishing \(F(3)\pm F(2)\pm F(2)=\mathbb R\). It is also described that \(F(3)F(2)F(2)\supseteq(-\infty,c]\cup[c,\infty)\) for a constant \(c\) and \(F(3)F(2)/F(2)=F(2)F(2)/F(3)=\mathbb R\backslash\{0\}\). NEWLINENEWLINENEWLINEPart II, see J. Number Theory 91, 187-205 (2001; Zbl 1030.11002)].