Totally real algebraic integers in small intervals

We show that there are irreducible monic polynomials having all roots in an interval of length close to 4. These are obtained by perturbing the coefficients of the respective Chebyshev polynomials. In particular, we obtain that our earlier lower bound for the house of totally real algebraic integers is sharp up to a logarithmic factor. Partially supported by the Lithuanian State Science and Studies Foundation. Published in Lietuvos Matematikos Rinkinys, Vol. 40, No. 3, pp. 307–312, July–September, 2000.