Upper and lower estimates of the number of algebraic points in short intervals

The distribution of algebraic numbers is quite complicated. Probably this is why they are rarely used as dense sets. Nevertheless, A. Baker and W. Schmidt proved in 1970 that the distribution of algebraic numbers still have some kind of uniformity on long intervals, which they called regularity. Recently many works have appeared addressing the problems concerning the lengths of the intervals on which real algebraic numbers have regularity property. It was discovered that for any integer $Q > 1$ there are intervals of length $0.5 Q^{-1}$, which don't contain algebraic numbers of any degree $n$ and of height $H(\alpha) \le Q$. At the same time it's possible to find such $c_0 = c_0(n)$ that for any $c > c_0$ algebraic numbers on any interval of length exceeding $c Q^{-1}$ have regularity property. Such "friendly"  to algebraic numbers intervals are intervals free of rational numbers with small denominators and algebraic numbers of small degree and height. In order to find algebraic numbers we build integral polynomials with small values on an interval and large height using Minkowski's linear forms theorem. It turns out that for "most"  points $x$ of an interval these polynomials have similar and nice characteristics (degree, height, module of polynomial value at point $x$). These characteristics are sufficient for building algebraic numbers on an interval. In this paper we prove existence of algebraic numbers of high degree on "very short" intervals.