AN INCIDENCE RESULT FOR WELL-SPACED ATOMS IN ALL DIMENSIONS

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.