## Functiones et Approximatio Commentarii Mathematici

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

Jerzy Kaczorowski

#### 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 $\theta$.

#### Article information

Source
Funct. Approx. Comment. Math., Volume 40, Number 1 (2009), 151-154.

Dates
First available in Project Euclid: 30 March 2009

https://projecteuclid.org/euclid.facm/1238418805

Digital Object Identifier
doi:10.7169/facm/1238418805

Mathematical Reviews number (MathSciNet)
MR2527636

Zentralblatt MATH identifier
1234.11152

#### Citation

Kaczorowski, Jerzy. A note on algebraic integers with prescribed factorization properties in short intervals. Funct. Approx. Comment. Math. 40 (2009), no. 1, 151--154. doi:10.7169/facm/1238418805. https://projecteuclid.org/euclid.facm/1238418805

#### References

• A. Geroldinger, F. Halter-Koch, Non-Unique Factorizations. Algebraic, Combinatorial and Analytic Theory, Pure and Applied Mathematics (Boca Raton), 278. Chapman & Hall/CRC, Boca Raton, FL, 2006.
• J. Kaczorowski, Irreducible algebraic integers in short intervals, to appear in Math. Ann.
• W. Narkiewicz, Elementary and analytic theory of algebraic numbers, Springer-Verlag, Berlin, Heidelberg, 2004.