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19
Erdős distance problem in vector spaces over finite fields
 Transactions of the American Mathematical Society
"... Abstract. We study the Erdös/Falconer distance problem in vector spaces over finite fields. Let Fq be a finite field with q elements and take E ⊂ F d q, d ≥ 2. We develop a Fourier analytic machinery, analogous to that developed by Mattila in the continuous case, for the study of distance sets in F ..."
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Cited by 34 (10 self)
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Abstract. We study the Erdös/Falconer distance problem in vector spaces over finite fields. Let Fq be a finite field with q elements and take E ⊂ F d q, d ≥ 2. We develop a Fourier analytic machinery, analogous to that developed by Mattila in the continuous case, for the study of distance sets in F d q to provide estimates for minimum cardinality of the distance set ∆(E) intermsofthe cardinality of E. Bounds for Gauss and Kloosterman sums play an important role in the proof. 1.
The number of different distances determined by n points in the plane
 J. Combin. Theory Ser. A
, 1984
"... A classical problem in combinatorial geometry is that of determining the minimum number f(n) of different distances determined by n points in the Euclidean plane. In 1952, L. Moser proved thatf(n)> n”‘/(2 fi) 1 and this has remained for 30 years as the best lower bound known for f(n). It is shown t ..."
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Cited by 22 (0 self)
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A classical problem in combinatorial geometry is that of determining the minimum number f(n) of different distances determined by n points in the Euclidean plane. In 1952, L. Moser proved thatf(n)> n”‘/(2 fi) 1 and this has remained for 30 years as the best lower bound known for f(n). It is shown that f(n)> cn “ ’ for some fixed constant c. I.
Distinct Distances in the Plane
, 2001
"... It is shown that every set of n points in the plane has an element from which there are at least cn 6/7 other elements at distinct distances, where c > 0 is a constant. This improves earlier results of Erdos, Moser, Beck, Chung, Szemeredi, Trotter, and Szekely. 1. ..."
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Cited by 16 (0 self)
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It is shown that every set of n points in the plane has an element from which there are at least cn 6/7 other elements at distinct distances, where c > 0 is a constant. This improves earlier results of Erdos, Moser, Beck, Chung, Szemeredi, Trotter, and Szekely. 1.
On Distinct Sums and Distinct Distances
, 2001
"... The paper [10] of J. Solymosi and Cs. Toth implicitly raised the following arithmetic problem. Consider n pairwise disjoint s element sets and form all s 2 n sums of pairs of elements of the same set. What is the minimum number of distinct sums one can get this way? This paper proves a lower bo ..."
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Cited by 13 (3 self)
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The paper [10] of J. Solymosi and Cs. Toth implicitly raised the following arithmetic problem. Consider n pairwise disjoint s element sets and form all s 2 n sums of pairs of elements of the same set. What is the minimum number of distinct sums one can get this way? This paper proves a lower bound on the number of distinct sums. As an application we improve the SolymosiToth bound on an old Erd}os problem: the number of distinct distances n points determine in the plane. Our bound also nds applications in other related results in discrete geometry. Our bounds are proven through an involved calculation of entropies of several random variables.
Incidences in Three Dimensions and Distinct Distances in the Plane (Extended Abstract)
, 2010
"... We first describe a reduction from the problem of lowerbounding the number of distinct distances determined by a set S of s points in the plane to an incidence problem between points and a certain class of helices (or parabolas) in three dimensions. We offer conjectures involving the new setup, but ..."
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Cited by 10 (5 self)
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We first describe a reduction from the problem of lowerbounding the number of distinct distances determined by a set S of s points in the plane to an incidence problem between points and a certain class of helices (or parabolas) in three dimensions. We offer conjectures involving the new setup, but are still unable to fully resolve them. Instead, we adapt the recent new algebraic analysis technique of Guth and Katz [9], as further developed by Elekes et al. [6], to obtain sharp bounds on the number of incidences between these helices or parabolas and points in R 3. Applying these bounds, we obtain, among several other results, the upper bound O(s 3) on the number of rotations (rigid motions) which map (at least) three points of S to three other points of S. In fact, we show that the number of such rotations which map at least k ≥ 3 points of S to k other points of S is close to O(s 3 /k 12/7). One of our unresolved conjectures is that this number is O(s 3 /k 2), for k ≥ 2. If true, it would imply the lower bound Ω(s / log s) on the number of distinct distances in the plane.
Distance sets of welldistributed planar point sets, Discrete Comput
 Geom
"... Abstract. We prove that a welldistributed subset of R 2 can have a distance set ∆ with #( ∆ ∩ [0,N]) ≤ CN 3/2−ɛ only if the distance is induced by a polygon K. Furthermore, if the above estimate holds with ɛ =1/2, then K can have only finitely many sides. ..."
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Cited by 5 (0 self)
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Abstract. We prove that a welldistributed subset of R 2 can have a distance set ∆ with #( ∆ ∩ [0,N]) ≤ CN 3/2−ɛ only if the distance is induced by a polygon K. Furthermore, if the above estimate holds with ɛ =1/2, then K can have only finitely many sides.
On Distinct Distances from a Vertex of a Convex Polygon
"... Given a set P of n points in convex position in the plane, we prove that there exists a point p 2 P such that the number of distinct distances from p is at least d(13n  6)/36e. The best previous bound, dn=3e, from 1952, is due to Leo Moser. ..."
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Cited by 4 (1 self)
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Given a set P of n points in convex position in the plane, we prove that there exists a point p 2 P such that the number of distinct distances from p is at least d(13n  6)/36e. The best previous bound, dn=3e, from 1952, is due to Leo Moser.
Lattices with Few Distances
 J. Number Theory
, 1991
"... this paper we prove all cases except n = 2 (for which see Smith [37]) of the following proposition. ..."
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Cited by 3 (0 self)
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this paper we prove all cases except n = 2 (for which see Smith [37]) of the following proposition.
On distance measures for welldistributed sets
, 2007
"... In this paper we investigate the Erdös/Falconer distance conjecture for a natural class of sets statistically, though not necessarily arithmetically, similar to a lattice. We prove a good upper bound for spherical means that have been classically used to study this problem. We conjecture that a majo ..."
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Cited by 3 (2 self)
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In this paper we investigate the Erdös/Falconer distance conjecture for a natural class of sets statistically, though not necessarily arithmetically, similar to a lattice. We prove a good upper bound for spherical means that have been classically used to study this problem. We conjecture that a majorant for the spherical means suffices to prove the distance conjecture(s) in this setting. For a class of nonEuclidean distances, we show that this generally cannot be achieved, at least in dimension two, by considering integer point distributions on convex curves and surfaces. In higher dimensions, we link this problem to the question about the existence of smooth wellcurved hypersurfaces that support many integer points.