Abstract. A classical improvement of the fundamental Minkowski--Blichfeldt theorem in geometry of numbers is the Siegel--Bombieri formula: if we know that for a bounded measurable set $A\subset{\msbm R}^n$ the algebraic difference $A-A$ contains no non-zero points of a full dimensional point lattice $\Lambda\subset {\msbm R}^n$, then $V(A)(d\Lambda -V(A))$ is equal (roughly speaking) to a non-negative Fourier series generated by $\Lambda $ and $A$, where $d\Lambda $ is the determinant of $\Lambda $ and $V(A)$ is the measure of $A$. In the paper both improvements and extensions (to arbitrary dimensional point lattices $L$) of this result are proved. These are achieved by using two principially new tools. First, a series of five consecutive inequalities between the cardinality of the set $(A-A)\cap L$ and the number 1, as well as exact conditions of equality in four of them, are proved. (These among other things show that the condition ``$(A-A)\cap L$ contains only the zero vector" can be satisfied only by quite special $A$.) Second, a special technique is worked out to extend the Fourier analytic methods of Siegel and Bombieri to point lattices $L$ of any dimensions.

AMS Subject Classification (1991): 11H16, 42B05

Received May 19, 1995 and in revised form March 5, 1996. (Registered under 5710/2009.)