MathDB

3

Part of 2018 China National Olympiad

Problems(1)

Simiultaneous Diophantine Approximation on Cube Roots

Source: China Mathematical Olympiad 2018 Q3

11/15/2017
Let qq be a positive integer which is not a perfect cube. Prove that there exists a positive constant CC such that for all natural numbers nn, one has {nq13}+{nq23}≥Cn−12\{ nq^{\frac{1}{3}} \} + \{ nq^{\frac{2}{3}} \} \geq Cn^{-\frac{1}{2}} where {x}\{ x \} denotes the fractional part of xx.
number theoryapproximationInequalityfractional partdiophantine approximation