Lilian Matthiesen, KTH Royal Institute of Technology
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.