Abstract: By ergodicity, any set of positive measure is expanded to a set of full measure under all translations in Z^d. But what happens if we restrict the translations to a specific line in Z^d? In this talk, we will explore this question and, using Furstenberg’s correspondence principle, demonstrate that the set of volumes of all simplices in Z^d formed by vertices of a positive-density set always contains an infinite arithmetic progression. This talk is based on joint works with M. Björklund (Chalmers) and S. Skinner (Sydney).