Sums of continued fractions to the nearest integer

Abstract. Let \(b\) be a positive integer. We prove that every real number can be written as sum of an integer and at most \(\lfloor {b+1\over 2}\rfloor\) continued fractions to the nearest integer each of which having partial quotients at least \(b\).

Key words and phrases. Continued fraction to the nearest integer, Hall's theorem.