Distributions Modulo One of Slowly Varying Functions of Primes

For a common class of slowly varying functions 𝑓, this note studies the empirical distributions modulo one of sequences (𝑓⁡(𝑝𝑛)) where 𝑝𝑛 denotes the 𝑛-th prime number. By means of a Kantorovich metric for probability measures on the circle, simple explicit bounds are established for the deviation of these empirical distributions from their limit set, a circle consisting of all rotated copies of a single exponential distribution. Despite their simplicity the established bounds often are sharp. They complement known results in the literature and give rise to several intriguing quantitative questions.