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The classical Dirichlet approximation theorem implies that, for any irrational number α, there exist infinitely many pairs of coprime integers a, q with q positive, satisfying |αq − a| < q^(−1). In other words, the distance of αq from the nearest integer, denoted by ∥αq∥, satisfies ∥αq∥ < q^(−1), for infinitely many positive integers q. A natural and interesting question is what happens if we restrict the denominators q to lie in subsets of the positive integers with particular arith-
metic structures. This question has led to an active line of research exploring the interplay between Diophantine approximation and the arithmetic properties of integers.
We studied this problem when the denominators q are restricted to certain sparse subsets S of positive integers, and on obtaining the best currently achievable exponent of q in place of −1. In this seminar, we will
discuss mainly three cases. Firstly, we consider the case where S is the set of Piatetski-Shapiro primes. These are primes of the forms [n^c] with 1 < c < 2 and forms a power saving sparse subset of the set of all primes. Secondly, we consider S to be the set of positive integers which can be written as sums of two squares. Lastly, we discuss the case where S is the set of smooth (or friable) numbers. |