# Continuous Incidence Theory and its Applications to Number Theory and Geometry

Friday, June 1, 2012 - 4:30pm - 5:00pm

Keller 3-180

It is a classical problem to study incidences between a nite number

of points and a nite number of geometric objects. In this presentation, we see that continuous incidence theory can be used to derive a number of results in geometry, geometric measure theory, and analytic number theory. The applications to geometry include a fractal variant of the regular value theorem. The applications to geometric measure theory include a generalization of Falconer distance problem in which we prove that a compact subset of R

described by certain restrictions. More specifically we consider the set

{(

where

ary and non-vanishing curvature.

The applications to Number Theory include counting integer lat-

tice points in the neighborhood of variable coecient families of sur-

faces.

of points and a nite number of geometric objects. In this presentation, we see that continuous incidence theory can be used to derive a number of results in geometry, geometric measure theory, and analytic number theory. The applications to geometry include a fractal variant of the regular value theorem. The applications to geometric measure theory include a generalization of Falconer distance problem in which we prove that a compact subset of R

^{d}of sufficiently large Hausdorff dimension determines a positive proportion of all (*k*+1)-configurationsdescribed by certain restrictions. More specifically we consider the set

{(

*x, x*)∈^{1},...,x^{k}*E*:^{k+1}*x - x*^{i}*=*_{B}*t*;1≤_{i}*i*≤*k*},where

*B*is any convex centrally symmetric body with a smooth bound-ary and non-vanishing curvature.

The applications to Number Theory include counting integer lat-

tice points in the neighborhood of variable coecient families of sur-

faces.

MSC Code:

11Hxx

Keywords: