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

Extract

The following paper is in two parts.

In Part 1 it is shown that Kronecker's theorem can be extended in the form

If 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 if

is 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 that

Where K, L depend on s, and N on s and the bounds of