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SUBCONVEX $L^p$
-SETS, WEYL’S INEQUALITY AND EQUIDISTRIBUTION
- Part of
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- Journal:
- Journal of the Institute of Mathematics of Jussieu / Volume 25 / Issue 5 / September 2026
- Published online by Cambridge University Press:
- 15 July 2026, pp. 2919-2959
- Print publication:
- September 2026
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- Article
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We examine sets $\mathscr A$
of natural numbers having the property that for some real number $p\in (0,2)$
, one has the subconvex bound $$\begin{align*}\int_0^1 \Big| \sum_{n\in \mathscr A\cap [1,N]}e(n\alpha)\Big|^p{\,\mathrm{d}} \alpha \ll N^{-1}|\mathscr A\cap [1,N]|^p. \end{align*}$$
We show that exponential sums over such sets satisfy inequalities analogous to Weyl’s inequality, and in many circumstances of the same strength as classical versions of Weyl’s bound. We also examine equidistribution of polynomials modulo $1$
in which the summands are restricted to these subconvex $L^p$
-sets. In addition, we describe applications to problems involving character sums and averages of arithmetic functions.
