Averages over hyperplanes, sum-product theory in vector spaces over finite fields and the Erdos-Falconer distance conjecture

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dc.creatorHart, Derrick
dc.creatorIosevich, Alex
dc.creatorKoh, Doowon
dc.creatorRudnev, Misha
dc.date2007-07-24
dc.date2007-07-31
dc.date.accessioned2026-07-07T08:20:51Z
dc.date.available2026-07-07T08:20:51Z
dc.descriptionWe prove a point-wise and average bound for the number of incidences between points and hyper-planes in vector spaces over finite fields. While our estimates are, in general, sharp, we observe an improvement for product sets and sets contained in a sphere. We use these incidence bounds to obtain significant improvements on the arithmetic problem of covering ${\mathbb F}_q$, the finite field with q elements, by $A \cdot A+... +A \cdot A$, where A is a subset ${\mathbb F}_q$ of sufficiently large size. We also use the incidence machinery we develope and arithmetic constructions to study the Erdos-Falconer distance conjecture in vector spaces over finite fields. We prove that the natural analog of the Euclidean Erdos-Falconer distance conjecture does not hold in this setting due to the influence of the arithmetic. On the positive side, we obtain good exponents for the Erdos -Falconer distance problem for subsets of the unit sphere in $\mathbb F_q^d$ and discuss their sharpness. This results in a reasonably complete description of the Erdos-Falconer distance problem in higher dimensional vector spaces over general finite fields.
dc.identifierhttps://arxiv.org/abs/0707.3473
dc.identifierhttp://arxiv.org/abs/0707.3473
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/135190
dc.subjectClassical Analysis and ODEs
dc.subjectCombinatorics
dc.titleAverages over hyperplanes, sum-product theory in vector spaces over finite fields and the Erdos-Falconer distance conjecture
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