Estimating the rate of successful Diophantine approximations

Cheng Shi
Cheng Shi

Cheng Shi is a rising senior (22’) from Shanghai, China. He studied in High School Affiliated to Shanghai Jiao Tong University before he came to Wesleyan. He is a mathematics and physics double major, and he is planning to pursue graduate studies in mathematics/applied mathematics after Wesleyan. In his spare time, he likes listening to music, reading, and singing.


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.


Your Name (Your Department)

Live Poster Session:
Thursday, July 29th 1:15-2:30pm EDT