Abstract
This article is the first one in a series of three. It contains concurrency results for sets of linear mappings of ℝ with few compositions and/or small image sets. The fine structure of such sets of mappings will be described in part II [3]. Those structure theorems can be considered as a first attempt to find Freiman-Ruzsa type results for a non-Abelian group. Part III [4] contains some geometric applications.
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References
Antal Balog andEndre Szemerédi: A statistical theorem of set addition,Combinatorica,14 (1994), 263–268.
József Beck: On the lattice property of the plane and some problems of Dirac, Motzkin and Erdős,Combinatorica,3 (3–4) (1983), 281–297.
György Elekes: On linear combinatorics II,Combinatorica, to appear.
György Elekes: On linear combinatorics III,Combinatorica, to appear.
Gregory A. Freiman:Foundations of a Structural Theory of Set Addition (in Russian), Kazan Gos. Ped. Inst., Kazan, 1966.
Gregory A. Freiman:Foundations of a Structural Theory of Set Addition, Translation of Mathematical Monographs vol. 37, Amer. Math. Soc., Providence, R.I., USA, 1973.
Tibor Gallai: Solution of problem 4065,Am. Math. Monthly,51 (1944), 169–171.
The Mathematics of Paul Erdős, Ron Graham and Jaroslav Nešetřil, eds. Springer, 1996.
J. Jackson:Rational Amusements for Winter Evenings. Longman Hurst Rees Orme and Brown, London, 1821.
Miklós Laczkovich andImre Z. Ruzsa:The Number of Homothetic Subsets, In: [8] Ron Graham and Jaroslav Nešetřil, eds., Springer, 1996.
Imre Z. Ruzsa: Arithmetical progressions and sum sets. Technical Report 91-51, DIMACS Center for Discrete Math. and Theoretical Comp. Sci., Rutgers University, 1991.
Imre Z. Ruzsa: Generalized arithmetic progressions and sum sets,Acta Math. Sci. Hung.,65 (1994), 379–388.
J. J. Sylvester: Problem 2473.Math. Questions from the Educational Times,8 (1867), 106–107.
J. J. Sylvester. Problem 2572.Math. Questions from the Educational Times,45 (1886), 127–128.
J. J. Sylvester: Problem 3019.Math. Questions from the Educational Times,45 (1886), 134.
J. J. Sylvester: Problem 11851.Math. Questions from the Educational Times,59, (1893), 98–99.
Endre Szemerédi: Regular partitions of graphs,Colloques Internationaux C.N.R.S.: Problèmes Combinatoires et Théorie des Graphes, No 260:399–401, 1978.
Endre Szemerédi andW. T. Trotter Jr: Extremal problems in Discrete Geometry,Combinatorica,3 (3–4) (1983), 381–392.
Vera T. Sós: personal communication.
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Dedicated to the memory of P. Erdős
Research partially supported by HU-NSF grants OTKA T014302 and T019367.