Open Sets Avoiding Integral Distances

We study open point sets in Euclidean spaces $$\mathbb{R }^d$$Rd without a pair of points an integral distance apart. By a result of Furstenberg, Katznelson, and Weiss such sets must be of Lebesgue upper density $$0$$0. We are interested in how large such sets can be in $$d$$d-dimensional volume. We determine the exact values for the maximum volumes of the sets in terms of the number of their connected components and dimension. Here techniques from diophantine approximation, algebra and the theory of convex bodies come into play. Our problem can be viewed as a counterpart to known problems on sets with pairwise rational or integral distances. It possibly opens a new research direction with strong links to topology and measure theory.

[1]  Anthony Quas,et al.  Distances in positive density sets in Rd , 2009, J. Comb. Theory, Ser. A.

[2]  Michael Frazier,et al.  Studies in Advanced Mathematics , 2004 .

[3]  Heinrich W. E. Jung.  Ueber die kleinste Kugel, die eine räumliche Figur einschliesst. , 1901 .

[4]  Elias M. Stein,et al.  Fourier Analysis: An Introduction , 2003 .

[5]  Two Optimization Problems for Convex Bodies in the n-dimensional Space Dedicated to the memory of Bernulf Weißbach , 2004 .

[6]  Paul Erdös.  Ulam, the man and the mathematician , 1985, J. Graph Theory.

[7]  H. Weyl.  Über die Gleichverteilung von Zahlen mod. Eins , 1916 .

[9]  Thomas Wolff,et al.  Recent work connected with the Kakeya problem , 2007 .

[10]  Sascha Kurz,et al.  There Are Integral Heptagons, no Three Points on a Line, no Four on a Circle , 2008, Discret. Comput. Geom..

[11]  R. D. Carmichael,et al.  Mathematical tables and formulae , 1958 .

[12]  Panos M. Pardalos,et al.  Lectures on Global Optimization , 2009 .

[13]  B. Bukh.  Measurable Sets With Excluded Distances , 2008 .

[14]  P. R. Scott,et al.  INEQUALITIES FOR CONVEX SETS , 2000 .

[15]  Kenneth Falconer,et al.  Unsolved Problems In Geometry , 1991 .

[16]  Ryan Schwartz,et al.  Rational Distances with Rational Angles , 2010 .

[17]  Murray S. Klamkin.  Unsolved Problems in Intuitive Mathematics, Vol. II, Unsolved Problems in Geometry (H. T. Croft, K. J. Falconer, and R. K Guy) , 1992, SIAM Rev..

[18]  József Solymosi,et al.  On a Question of Erdős and Ulam , 2008, Discret. Comput. Geom..

[19]  R. Gardner.  Geometric Tomography: Parallel X-rays of planar convex bodies , 2006 .

[20]  Eric Schmutz,et al.  Rational points on the unit sphere , 2008 .

[21]  Frank de Zeeuw.  An Algebraic View of Discrete Geometry , 2011 .

[22]  B. V. Dekster.  The Jung Theorem in metric spaces of curvature bounded above , 1997 .

[23]  Péter Komjáth,et al.  Problems And Theorems In Classical Set Theory , 2006 .

[24]  P. McMullen.  GEOMETRIC TOMOGRAPHY (Encyclopedia of Mathematics and its Applications 58) , 1997 .

[25]  J. M. H. Olmsted.  Rational Values of Trigonometric Functions , 1945 .

[26]  F. M. D. O. Filho,et al.  Fourier analysis, linear programming, and densities of distance avoiding sets in R^n , 2008, 0808.1822.

[27]  Henry B. Mann,et al.  On linear relations between roots of unity , 1965 .

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

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

[30]  P. Flajolet,et al.  Analytic Combinatorics: RANDOM STRUCTURES , 2009 .

[31]  Larry Guth,et al.  On the Erdos distinct distance problem in the plane , 2010, 1011.4105.

[32]  Fernando Mário Oliveira Filho,et al.  Lower Bounds for Measurable Chromatic Numbers , 2008, 0801.1059.

[33]  Y. Bugeaud.  Distribution Modulo One and Diophantine Approximation: References , 2012 .

[34]  S. Ulam.  A collection of mathematical problems , 1960 .

[35]  L. Evans.  Measure theory and fine properties of functions , 1992 .

[36]  R. Graham,et al.  Are There n + 2 Points in E n With Odd Integral Distances? , 1974 .

[37]  PETER BRASS,et al.  A Lower Bound for Lebesgue's Universal Cover Problem , 2005, Int. J. Comput. Geom. Appl..

[38]  B. Weiss,et al.  Ergodic Theory and Configurations in Sets of Positive Density , 1990 .

[39]  Philippe Flajolet,et al.  Analytic Combinatorics , 2009 .

[40]  L. Piepmeyer,et al.  The maximum number of odd integral distances between points in the plane , 1996, Discret. Comput. Geom..

[41]  Pierre Hansen,et al.  Extremal Problems for Convex Polygons - An Update , 2007 .

[42]  Vojtěch Rödl,et al.  Mathematics of Ramsey Theory , 1991 .

[43]  H. T. Croft.  Unsolved Problems in Geometry Unsolved Problems in Intuitive Mathematics , 1991 .