Results on sums of continued fractions
James L. Hlavka · Transactions of the American Mathematical Society · 1975
Let F ( m ) F(m) be the (Cantor) set of infinite continued fractions with partial quotients no greater than m and let F ( m ) + F ( n ) = { α + β : α ∈ F ( m ) , β ∈ F ( n ) } F(m) + F(n) = \{ \alpha + \beta :\alpha \in F(m),\beta \in F(n)\} . We show that F ( 3 ) + F ( 4 ) F(3) + F(4) is an interval of length 1.14 ... so every real number is the sum of an integer, an element of F ( 3 ) F(3) and an element of F ( 4 ) F(4) . Similar results are given for F ( 2 ) + F ( 7 ) , F ( 2 ) + F ( 2 ) + F ( 4 ) , F ( 2 ) + F ( 3 ) + F ( 3 ) F(2) + F(7),F(2) + F(2) + F(4),F(2) + F(3) + F(3) and