An extended form of Kronecker's theorem with an application which shows that Burgers' theorem on adiabatic invariants is statistically true for an assembly

Abstract

The following paper is in two parts.In Part 1 it is shown that Kronecker's theorem can be extended in the formIf is greater than ηfor all sets of integers l1, l2, … l3 less in absolute value than K/σ and not all zero, l also being an integer, then, for any x1, x2 … x3, an integer q less than l/(ησ8) can be found such that qv1–x1, qv2–x2…qv3–x4 all differ from integers by less than σ;. K, L depend only on s.It is an immediate corollary that ifis greator than for all sets of integers l1, l2… l3, l less in absolute value than K/σ and not all zero while F(v1, v2… v3), l less in absolute value than K/σ and not all zero while F(v1, v2… v3, v) is periodic period 1, in v1, v2 … v3,v, then T can be found between 0 and such thatWhere K, L depend on s, and N on s and the bounds of