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AN INCIDENCE RESULT FOR WELL-SPACED ATOMS IN ALL DIMENSIONS
Part of
- Linear incidence geometry
- Geometry
PETER J. BRADSHAW
Journal:

Journal of the Australian Mathematical Society / Volume 115 / Issue 1 / August 2023

Published online by Cambridge University Press:

20 February 2023, pp. 58-72

Print publication:

August 2023
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We 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.