adicapplicationsapproximableapproximationclayconjecturecontinuedcounterexamplescountingdimensiondiophantinedistributiondistributionseffectiveequidistributioneveryexpansionfractiongapgroupshausdorffhigherhomogeneousinstitutelatticelatticeslecturelielinearlittlewoodmanifoldsmapsmathematicsmetricmultiplicativeorbitspointspotentialpropertiespropertyquadraticrandomratnerscholarseniorsimultaneousspacesspectralsumssurfacestheoremwalks