Abstract :
Let ξ be an irrational number with simple continued fraction expansion ξ = [a0; a1, a2, ..., ai, ...]. Let the ith convergent pi/qi = [a0; a1, a2, ..., ai]. Let μ = [0; an + 2, an + 3, ...] − [0; an, an − 1, ..., a1]. In this note, we prove that among three consecutive convergents pi/qi (i = n − 1, n, n + 1), at least one satisfies ξ − pi/qi < ([formula]), and at least one satisfies ξ − pi/qi > ([formula]). The results are best possible.
