# Distinct Angles and Angle Chains in Three Dimensions

TL;DR: In this paper , Fleischmann, Konyagin, Miller, Palsson, Pesikoff, and Wolf provide bounds on the minimum number of distinct angles in general position in three dimensions.

Abstract: In 1946, Erd\H{o}s posed the distinct distance problem, which seeks to find
the minimum number of distinct distances between pairs of points selected from
any configuration of $n$ points in the plane. The problem has since been
explored along with many variants, including ones that extend it into higher
dimensions. Less studied but no less intriguing is Erd\H{o}s' distinct angle
problem, which seeks to find point configurations in the plane that minimize
the number of distinct angles. In their recent paper "Distinct Angles in
General Position," Fleischmann, Konyagin, Miller, Palsson, Pesikoff, and Wolf
use a logarithmic spiral to establish an upper bound of $O(n^2)$ on the minimum
number of distinct angles in the plane in general position, which prohibits
three points on any line or four on any circle.
We consider the question of distinct angles in three dimensions and provide
bounds on the minimum number of distinct angles in general position in this
setting. We focus on pinned variants of the question, and we examine explicit
constructions of point configurations in $\mathbb{R}^3$ which use
self-similarity to minimize the number of distinct angles. Furthermore, we
study a variant of the distinct angles question regarding distinct angle chains
and provide bounds on the minimum number of distinct chains in $\mathbb{R}^2$
and $\mathbb{R}^3$.

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TL;DR: In this paper, the sets of distances of n points have been studied in the setting of sets of points, and the American Mathematical Monthly: Vol. 53, No. 5, pp. 248-250.

Abstract: (1946). On Sets of Distances of n Points. The American Mathematical Monthly: Vol. 53, No. 5, pp. 248-250.

592 citations

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TL;DR: In this paper, it was shown that a set of points in R 2 has at least c N log N distinct distances, thus obtaining the sharp exponent in a problem of Erd} os.

Abstract: In this paper, we prove that a set of N points in R 2 has at least c N log N distinct distances, thus obtaining the sharp exponent in a problem of Erd} os. We follow the setup of Elekes and Sharir which, in the spirit of the Erlangen program, allows us to study the problem in the group of rigid motions of the plane. This converts the problem to one of point-line incidences in space. We introduce two new ideas in our proof. In order to control points where many lines are incident, we create a cell decomposition using the polynomial ham sandwich theorem. This creates a dichotomy: either most of the points are in the interiors of the cells, in which case we immediately get sharp results or, alternatively, the points lie on the walls of the cells, in which case they are in the zero-set of a polynomial of suprisingly low degree, and we may apply the algebraic method. In order to control points incident to only two lines, we use the ecnode polynomial of the Rev. George Salmon to conclude that most of the lines lie on a ruled surface. Then we use the geometry of ruled surfaces to complete the proof.

439 citations

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TL;DR: Upper and lower bounds for extremal problems defined for arrangements of lines, circles, spheres, and alike are presented and it is proved that the maximum number of edges boundingm cells in an arrangement ofn lines is Θ(m2/3n 2/3 +n), and that it isO(m3/2β(m) forn unit-circles.

Abstract: We present upper and lower bounds for extremal problems defined for arrangements of lines, circles, spheres, and alike. For example, we prove that the maximum number of edges boundingm cells in an arrangement ofn lines is ?(m2/3n2/3 +n), and that it isO(m2/3n2/3s(n) +n) forn unit-circles, wheres(n) (and laters(m, n)) is a function that depends on the inverse of Ackermann's function and grows extremely slowly. If we replace unit-circles by circles of arbitrary radii the upper bound goes up toO(m3/5n4/5s(n) +n). The same bounds (without thes(n)-terms) hold for the maximum sum of degrees ofm vertices. In the case of vertex degrees in arrangements of lines and of unit-circles our bounds match previous results, but our proofs are considerably simpler than the previous ones. The maximum sum of degrees ofm vertices in an arrangement ofn spheres in three dimensions isO(m4/7n9/7s(m, n) +n2), in general, andO(m3/4n3/4s(m, n) +n) if no three spheres intersect in a common circle. The latter bound implies that the maximum number of unit-distances amongm points in three dimensions isO(m3/2s(m)) which improves the best previous upper bound on this problem. Applications of our results to other distance problems are also given.

362 citations

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01 Jan 2004

TL;DR: On the complexity of the linkage reconfiguration problem by H. Alt, C. Knauer, G. Rote, and S. Toth as mentioned in this paper, a new entropy inequality for the Erdos distance problem by N. Woyczynski.

Abstract: On the complexity of the linkage reconfiguration problem by H. Alt, C. Knauer, G. Rote, and S. Whitesides Falconer conjecture, spherical averages and discrete analogs by G. Arutyunyants and A. Iosevich Turan-type extremal problems for convex geometric hypergraphs by P. Brass The thrackle conjecture for $K_5$ and $K_{3,3}$ by G. Cairns, M. McIntyre, and Y. Nikolayevsky Three-dimensional grid drawings with sub-quadratic volume by V. Dujmovic and D. R. Wood On a coloring problem for the integer grid by A. Dumitrescu and R. Radoicic Separating thickness from geometric thickness by D. Eppstein Direction trees in centered polygons by R. E. Jamison Path coverings of two sets of points in the plane by A. Kaneko, M. Kano, and K. Suzuki Length of sums in a Minkowski space by G. O. H. Katona, R. Mayer, and W. A. Woyczynski A new entropy inequality for the Erdos distance problem by N. H. Katz and G. Tardos Coloring intersection graphs of geometric figures with a given clique number by A. Kostochka Convex quadrilaterals and $k$-sets by L. Lovasz, K. Vesztergombi, U. Wagner, and E. Welzl Distance graphs and rigidity by H. Maehara A Ramsey property of planar graphs by J. Nesetril, J. Solymosi, and P. Valtr A generalization of quasi-planarity by J. Pach, R. Radoicic, and G. Toth Geometric incidences by J. Pach and M. Sharir Large sets must have either a $k$-edge or a $(k+2)$-edge by M. A. Perles and R. Pinchasi Topological graphs with no self-intersecting cycle of length 4 by R. Pinchasi and R. Radoicic A problem on restricted sumsets by I. Z. Ruzsa The gap between crossing numbers and convex crossing numbers by F. Shahrokhi, O. Sykora, L. A. Szekely, and I. Vrto Distinct distances in high dimensional homogeneous sets by J. Solymosi and V. Vu The biplanar crossing number of the random graph by J. Spencer The unit distance problem on spheres by K. J. Swanepoel and P. Valtr Short proof for a theorem of Pach, Spencer, and Toth by L. Szekely.

178 citations

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