Almost periodic functions and an analytical method of solving the number partitioning problem

In the present paper, we study the limit sets of the almost periodic functions $f(x)$. It is interesting that the values $r=\inf|f(x)|$ and $R=\sup|f(x)|$ may be expressed in the exact form. We show that the ring $r\leq |z|\leq R$ is the limit set of the almost periodic function $f(x)$ (under some natural conditions on $f$). The exact expression for $r$ coincides with the well known partition problem formula and gives a new analytical method of solving the corresponding partition problem. Several interesting examples are considered. For instance, in the case of the five numbers, the well-known Karmarkar--Karp algorithm gives the value $m=2$ as the solution of the partition problem in our example, and our method gives the correct answer $m=0.$ The figures presented in Appendix illustrate our results.