A note on algebraic integers with prescribed factorization properties in short intervals

Abstract

We study the distribution of algebraic integers with prescribed factorization properties in short intervals and prove that for a large class of such numbers from a fixed algebraic number field $K$ with a non-trivial class group, every interval of the form $(x, x+x^{\theta})$ with a fixed $\theta >1/2$ contains absolute value of the norm of such algebraic integer provided $x\geq x_0$. The constant $x_0$ effectively depends on $K$ and $\theta$.