The values for which is close to 1 (say in an interval ) are some what regular: implies that there exists an integer k(n) such that where . As , thus we can safely say that for small enough, If and since and are both in , then we have the inequality where is some integer k. Since has a finite irrationality measure, we know that there is a finite real constant such that for any integers large enough, By picking small enough we can forget about the finite number of exceptions to the inequality, and we get . Thus for some constant A. Therefore, we have a guarantee on the lengh of the gaps between equally problematic terms, and we know how this length grows as gets smaller We can get a lower bound for the first problematic term using the irrationality measure as well : from , we get that for small enough, , and then for some constant B. Therefore, there exists a constant C such that forall small enough, the k-th integer n such that is greater than
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