Seminar in Algebraic Geometry and Number Theory

Abstract: We say an integer is $[y',y]$-smooth if all its prime factors are in the interval $[y',y]$. Suppose that $K'\geq1$ is a large integer and $y'=\log^{K'}N$. In joint work with Lilian Matthiesen we prove that $[y',y]$-smooth numbers are orthogonal to nilsequences (generalized polynomial phase functions) as long as $\log^K N\leq y\leq N$ and $K/K'$ is sufficiently large. As an application, when the set of $[y',y]$-smooth numbers is rather dense, we can also count the number of solutions of linear equations in $[y',y]$-smooth numbers.