On Some Diophantine Inequalities Involving the Exponential Function

It is well known that for any real number θ there are infinitely many positive integersnsuch that Here ||a|| denotes the distance of a from the nearest integer, taken positively. Indeed, since ||a|| < 1, this implies more generally that ifθ1,θ2, . . . ,θkare any real numbers, then there are infinitely many positive integersnsuch that