Diophantine approximation studies how well real numbers can be approximated by rational numbers. Previous theorems, such as Khintchine’s theorem and the Duffin-Schaeffer conjecture, have implied that the inequalities |x-p/q| < a/q2, with p/q being Farey fractions, can accept infinitely many solutions for almost every x within the interval [0,1]. Our project studies a pertinent and interesting extension of this statement.
We intended to find the number of solutions of such inequalities with the denominator q < Q for a fixed Q. Our conjecture is that, under the limit when Q goes to infinity, the ratio between the number of solutions and the summation of the lengths of the intervals [p/q – a/q2, p/q + a/q2] converges to 1. Though we haven’t proven it yet, we’ve managed to provide an upper bound and a lower bound for this ratio using the method of continued fractions when a < 1/2. The fact that these bounds are so close to 1 further supported the validity of our conjecture. Using a probabilistic argument, we’ve also proven that the limit is indeed 1 when the fraction p/q only takes prime denominators.
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Thursday, July 29th 1:15-2:30pm EDT