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In the present paper, we study the limit sets of the almost periodicfunctions $f(x)$. 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 somenatural conditions on $f$). The exact expression for $r$ coincides with thewell known partition problem formula and gives a new analytical method ofsolving the corresponding partition problem. Several interesting examples areconsidered. For instance, in the case of the five numbers, the well-knownKarmarkar--Karp algorithm gives the value $m=2$ as the solution of thepartition problem in our example, and our method gives the correct answer$m=0.$ The figures presented in Appendix illustrate our results.

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