On the limit shape of elements of an arithmetic semigroup with an exponentially growing counting function of basis elements

Abstract

We consider an arithmetic semigroup with exponential growth of the counting function of abstract primes. The Bose–Einstein statistics provides the most probable mean occupation numbers in the sense that large deviations of a sum of occupation numbers from the corresponding sum for the Bose–Einstein statistics have small probabilities. The probabilities of large deviations are estimated.