Distinct distances between a collinear set and an arbitrary set of points

Abstract We consider the number of distinct distances between two finite sets of points in R k , for any constant dimension k ≥ 2 , where one set P 1 consists of n points on a line l , and the other set P 2 consists of m arbitrary points, such that no hyperplane orthogonal to l and no hypercylinder having l as its axis contains more than O ( 1 ) points of P 2 . The number of distinct distances between P 1 and P 2 is then Ω min n 2 ∕ 3 m 2 ∕ 3 , n 10 ∕ 11 m 4 ∕ 11 log 2 ∕ 11 m , n 2 , m 2 . Without the assumption on P 2 , there exist sets P 1 , P 2 as above, with only O ( m + n ) distinct distances between them.

[1]  G. Elekes,et al.  A Note on the Number of Distinct Distances , 1999 .

[2]  Paul Erdös.  On some problems of elementary and combinatorial geometry , 1975 .

[3]  Lajos Rónyai,et al.  A Combinatorial Problem on Polynomials and Rational Functions , 2000, J. Comb. Theory, Ser. A.

[4]  P. Erdös.  On Sets of Distances of n Points , 1946 .

[5]  János Pach,et al.  Research problems in discrete geometry , 2005 .

[6]  Marcos Charalambides,et al.  Distinct Distances on Curves Via Rigidity , 2013, Discret. Comput. Geom..

[7]  János Pach,et al.  Distinct Distances on Algebraic Curves in the Plane , 2017, Comb. Probab. Comput..

[8]  Orit E. Raz,et al.  A note on distinct distances , 2016, Combinatorics, Probability and Computing.

[9]  Joshua Zahl,et al.  Few distinct distances implies no heavy lines or circles , 2016, Comb..

[10]  Ruixiang Zhang,et al.  Bounds of incidences between points and algebraic curves , 2013, 1308.0861.

[11]  Micha Sharir,et al.  Distinct distances on two lines , 2013, J. Comb. Theory, Ser. A.

[12]  Orit E. Micha József Raz,et al.  Polynomials vanishing on grids: The Elekes-Rónyai problem revisited , 2014 .

[13]  Micha Sharir,et al.  Lenses in arrangements of pseudo-circles and their applications , 2004, JACM.

[14]  Joe W. Harris,et al.  Algebraic Geometry: A First Course , 1995 .

[15]  Orit E. Raz,et al.  Polynomials vanishing on grids: The Elekes-Rónyai problem revisited , 2014, SoCG.

[16]  Orit E. Raz,et al.  Sets with few distinct distances do not have heavy lines , 2015, Discret. Math..

[17]  Gábor Tardos,et al.  Intersection reverse sequences and geometric applications , 2004, J. Comb. Theory, Ser. A.

[18]  L. Guth,et al.  On the Erdős distinct distances problem in the plane , 2015 .