On the chromatic numbers of integer and rational lattices

In this paper, we give new upper bounds for the chromatic numbers for integer lattices and some rational spaces and other lattices. In particular, we have proved that for any concrete integer number $d$, the chromatic number of $\mathbb Z^n$ with critical distance $\sqrt{2d}$ has a polynomial growth in $n$ with exponent less than or equal to $d$ (sometimes this estimate is sharp). The same statement is true not only in the Euclidean norm, but also in any $l_p$ norm. Moreover, we have given concrete estimates for some small dimensions as well as upper bounds for the chromatic number of $\mathbb Q_p^n$, where by $\mathbb Q_p$ we mean the ring of all rational numbers having denominators not divisible by some prime numbers.