On distinct perpendicular bisectors and pinned distances in finite fields
Copyright policy )On distinct perpendicular bisectors and pinned distances in finite fields
Given a set of points $P \subset \mathbb F_q^2$ such that $|P|\geq q^{3/2}$ it is established that $|P|$ determines $Ω(q^2)$ distinct perpendicular bisectors. It is also proven that, if $|P| \geq q^{4/3}$, then for a positive proportion of points $a \in P$, we have $$|\{\| a- b\|: b \in P\}|=Ω(q),$$ where $\|a- b\|$ is the distance between points $a$ and $b$. The latter result represents an improvement on a result of Chapman et al. (arxiv:0903.4218).
- Wuhan University China (People's Republic of)
- Pennsylvania State University United States
- Rutgers, The State University of New Jersey United States
- Rutgers University New Brunswick United States
rigid motions, Mathematics - Number Theory, pinned distances, perpendicular bisectors, 52C10, Erdős problems and related topics of discrete geometry, Arrangements of points, flats, hyperplanes (aspects of discrete geometry), Structure theory for finite fields and commutative rings (number-theoretic aspects), Arithmetic combinatorics; higher degree uniformity, expander mixing lemma, FOS: Mathematics, Mathematics - Combinatorics, Exponential sums, isosceles triangles, Combinatorics (math.CO), Number Theory (math.NT), finite fields
rigid motions, Mathematics - Number Theory, pinned distances, perpendicular bisectors, 52C10, Erdős problems and related topics of discrete geometry, Arrangements of points, flats, hyperplanes (aspects of discrete geometry), Structure theory for finite fields and commutative rings (number-theoretic aspects), Arithmetic combinatorics; higher degree uniformity, expander mixing lemma, FOS: Mathematics, Mathematics - Combinatorics, Exponential sums, isosceles triangles, Combinatorics (math.CO), Number Theory (math.NT), finite fields
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