Abstract
Write fk(α;X)=∑x⩽Xe(α1x+⋯+αkxk) (k⩾3). We show that there is a set B⊆[0,1)k−2 of full measure with the property that whenever (α2,…,αk−1)∈B and X is sufficiently large, then
sup(α1,αk)∈[0,1)2|fk(α;X)|⩽X1/2+δk,
where
δk=min{1330,42k−1}.
For k⩾5, this improves on work of Flaminio and Forni, in which a Diophantine condition is imposed on αk, and the exponent of X is 1−2/(3k(k−1)).
| Original language | English |
|---|---|
| Pages (from-to) | 2632-2646 |
| Number of pages | 15 |
| Journal | International Mathematics Research Notices |
| Volume | 2016 |
| Issue number | 9 |
| Early online date | 24 Jul 2015 |
| DOIs | |
| Publication status | Published - 10 May 2016 |