A note on algebraic integers with prescribed factorization properties in short intervals (Q1032740)

Let \(K\) be an algebraic number field and \(O_K\) its ring of integers. If \(X\) is a class of ideals in \(K\), then for \(a\in O_K\) one denotes by \(\Omega_X(a)\) the number of prime ideal factors of the ideal \(aO_K\) (counted with multiplicities). A set of ideals \(\mathcal A\) is called regular, if there exist ideal classes \(X_1,X_2,\dots,X_n\) and integers \(c_1,c_2,\dots,c_n\) such that \(\mathcal A\) contains all \(a\in O_K\) satisfying \(\Omega_{X_i}=c_i\) for \(i=1,2,\dots,n\). The maximal possible sum \(\sum_{i=1}^nc_i\) is called the size of \(\mathcal A\). The author shows that if \(\mathcal A\) is of size \(\geq3\), then for every \(\theta>1/2\) and sufficiently large \(x\) there exists \(a\in \mathcal A\) with \(x<|N(a)|<x+x^\theta\). The method of the proof is a modification of the argument used by the author in a previous paper [Math. Ann. 345, No. 2, 297--305 (2009; Zbl 1233.11114)], where he obtained this assertion in the case when \(\mathcal A\) is the set of all irreducible elements of \(O_K\).