I suspect the following statement is true and I can use it in my work if it is true. However I am not a number theorist and I could not prove it myself. I was wondering if this is known to number theorists. 


Statement: For each positive real number $\alpha$ there exist a natural number $N$ such that, for every $n \geq N$ each of the intervals $[n^{\alpha},2n^{\alpha}) , [2n^{\alpha},3n^{\alpha}), ...$ contains at least one natural number that has no common factor with $n$.



Existence of range of numbers containing a coprime to given n
This is tightly related to the *Jacobsthal function* defined to be the smallest integer $j(n)$ such that any segment of $j(n)$ consecutive integers contains an integer co-prime with $n$. Iwaniec ("On the problem of Jacobsthal", *Demonstratio Math.* **11**, 225–231, 1978) proved that $j(n)=O(\log^2(n))$; this implies your statement.


