Abstract
AbstractWe prove an incidence result counting the k-rich
$\delta $
-tubes induced by a well-spaced set of
$\delta $
-atoms. Our result coincides with the bound that would be heuristically predicted by the Szemerédi–Trotter theorem and holds in all dimensions
$d \geq 2$
. We also prove an analogue of Beck’s theorem for
$\delta $
-atoms and
$\delta $
-tubes as an application of our result.
Publisher
Cambridge University Press (CUP)