Activities This Week

The Dickmann’s law was discovered in the number theory (statistics of the natural numbers with a small prime factors). The Derrida’s model of the random energies demonstrated the physical phase transitions of the second type. These models are from the completely different areas, however they have the same background and many similarities. The talk will contain the discussion of such similarities and the numerous applications, in particular, to the cell growth model.

Given a set S of positive measure on the unit circle, a set of integers K is an interpolation set (IS) for S if for any data ${c(k)}$ in $l^2(K)$ there exists a function $f$ in $L^2(S)$ such that its Fourier coefficients satisfy $\hat{f}(k)=c(k)$ for all k in K. In the talk I will discuss the relationship between the concept of IS and the existence arithmetic structure in the set K, I will focus primarily on the case where K contains arbitrarily long arithmetic progressions with specified lengths and step sizes. Multidimensional analogue and recent developments on this subject will also be considered. This talk is based in part on joint work with Alexander Olevskii.