Equidistribution and $L^p$-sets for $p<2$

Abstract: We investigate subsets $\mathcal A$ of the natural numbers having the property that, for some positive number $p<2$, one has \[ \int_0^1 \Bigl| \sum_{n\in \mathcal A\cap [1,N]}e(n\alpha)\Bigr|^p\,{\rm d}\alpha \ll |\mathcal A\cap [1,N]|^pN^{\varepsilon-1}. \] Examples of such sets include (but are not restricted to) the squarefree, or more generally, the $r$-free numbers. For polynomials $\psi(x;\boldsymbol\alpha)=\alpha _kx^k+\ldots +\alpha_1x$, having coefficients $\alpha_i$ satisfying suitable irrationality conditions, we show that the sequence $(\psi(n;\boldsymbol\alpha))_{n\in \mathcal A}$ is equidistributed modulo $1$.

Mathematics

Audience: researchers in the topic

Combinatorial and additive number theory (CANT 2025)