Distributional properties of smooth numbers: Smooth numbers are orthogonal to nilsequences

Abstract: An integer is called y-smooth if all of its prime factors are of size at most y. The y-smooth numbers below x form a subset of the integers below x which is, in general, sparse but is known to enjoy good equidistribution properties in progressions and short intervals. Distributional properties of y-smooth numbers found striking applications in, for instance, integer factorisation algorithms or in work of Vaughan and Wooley on improving bounds in Waring's problem. In this talk I will discuss joint work with Mengdi Wang which considers some finer aspects of the distribution of y-smooth numbers. More precisely, we show for a very large range of the parameter y that y-smooth number are (in a certain sense) discorrelated with "nilsequences". Through work of Green, Tao and Ziegler, our result is closely related to the Diophantine problem of studying solutions to certain systems of linear equations in the set of y-smooth numbers.

number theory

Audience: researchers in the topic

Around Frobenius Distributions and Related Topics IV

Series comments: Registration is free, but all participants are required to register on the conference website.