Abstract
Ulam asked in 1945 if there is an everywhere dense rational set, i.e., 1 a point set in the plane with all its pairwise distances rational. Erdős conjectured that if a set S has a dense rational subset, then S should be very special. The only known types of examples of sets with dense (or even just infinite) rational subsets are lines and circles. In this paper we prove Erdős’ conjecture for algebraic curves by showing that no irreducible algebraic curve other than a line or a circle contains an infinite rational set.
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Anning, N.H., Erdős, P.: Integral distances. Bull. Am. Math. Soc. 51, 598–600 (1945)
Brass, P., Moser, W., Pach, J.: Research Problems in Discrete Geometry, 1st edn. Springer, Berlin (2005). XII, 499 p.
Campbell, G.: Points on y=x 2 at rational distance. Math. Comput. 73, 2093–2108 (2004)
Choudhry, A.: Points at rational distances on a parabola. Rocky Mt. J. Math. 36(2), 413–424 (2006)
Erdős, P.: Integral distances. Bull. Am. Math. Soc. 51, 996 (1945)
Erdős, P.: Verchu niakoy geometritchesky zadatchy. Fiz.-Mat. Spis. Bŭlgar. Akad. Nauk 5(38), 205–212 (1962). (On some geometric problems, in Bulgarian)
Erdős, P.: On some problems of elementary and combinatorial geometry. Ann. Mat. Pura Appl. (IV) CIII, 99–108 (1975)
Erdős, P.: Néhány elemi geometriai problémáról. Középisk Mat. Lapok 61, 49–54 (1980). (On some problems in elementary geometry, in Hungarian)
Erdős, P.: Combinatorial problems in geometry. Math. Chron. 12, 35–54 (1983)
Erdős, P.: Ulam, the man and the mathematician. J. Graph Theory 9(4), 445–449 (1985) Also appears in Creation Math. 19 (1986), 13–16
Erdős, P.: Some combinatorial and metric problems in geometry. In: Colloquia Mathematica Societatis János Bolyai, vol. 48, pp. 167–177. Intuitive Geometry, Siófok (1985)
Erdős, P., Purdy, G.B.: Extremal problems in combinatorial geometry. In: Graham, R.L., Grötschel, M., Lovász, L. (eds.) Handbook of Combinatorics, pp. 809–875. Elsevier, Amsterdam (1995)
Faltings, G.: Endlichkeitssätze für abelsche Varietäten über Zahlkörpern. Invent. Math. 73(3), 349–366 (1983). (Finiteness theorems for abelian varieties over number fields)
Guy, R.: Unsolved Problems in Number Theory, 3rd edn. Problem Books in Mathematics Subseries: Unsolved Problems in Intuitive Mathematics, vol. 1. Springer, Berlin (2004). XVIII, 438 p.
Harborth, H., Kemnitz, A., Möller, M.: An upper bound for the minimum diameter of integral point sets. Discrete Comput. Geom. 9(4), 427–432 (1993)
Huff, G.B.: Diophantine problems in geometry and elliptic ternary forms. Duke Math. J. 15, 443–453 (1948)
Kemnitz, A.: Punktmengen mit ganzzahligen Abständen. Habilitationsschrift, TU Braunschweig (1988)
Kreisel, T., Kurz, S.: There are integral heptagons, no three points on a line, no four on a circle. Discrete Comput. Geom. 39(4), 786–790 (2008)
Peeples, W.D. Jr.: Elliptic curves and rational distance sets. Proc. Am. Math. Soc. 5, 29–33 (1954)
Silverman, J.: The Arithmetic of Elliptic Curves. Springer, Berlin (1986)
Solymosi, J.: Note on integral distances. Discrete Comput. Geom. 30(2), 337–342 (2003)
Ulam, S.M.: A Collection of Mathematical Problems. Interscience Tracts in Pure and Applied Mathematics, vol. 8. Interscience, New York (1960). XIII, 150 p.
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The first author was supported by NSERC and OTKA grants and by Sloan Research Fellowship.
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Solymosi, J., de Zeeuw, F. On a Question of Erdős and Ulam. Discrete Comput Geom 43, 393–401 (2010). https://doi.org/10.1007/s00454-009-9179-x
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DOI: https://doi.org/10.1007/s00454-009-9179-x