Source: Real Anal. Exchange
Volume 36, Number 1
We construct a $d$ Hausdorff dimensional compact set in $\R^d$ that does not contain the vertices of any parallelogram. We also prove that for any given triangle ($3$ given points in the plane) there exists a compact set in $\R^2$ of Hausdorff dimension $2$ that does not contain any similar copy of the triangle. On the other hand, we show that the set of the $3$-point patterns of a $1$-dimensional compact set of $\R$ is dense.
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P. Erdős, Remarks on some problems in number theory, Math. Balkanica 4 (1974), 197-202.
Mathematical Reviews (MathSciNet): MR429704
K. Falconer, Fractal geometry, John Wiley & Sons (1990).
W. T. Gowers, A new proof of Szemerédi's theorem, Geom. Funct. Anal. 11 (2001), 465-588.
B. Green, T. Tao, New bounds for Szemerédi's theorem II.: A new bound for $r_4(N)$, Analytic number theory: essays in honour of Klaus Roth, W. W. L. Chen, W. T. Gowers, H. Halberstam, W. M. Schmidt, R. C. Vaughan, eds, Cambridge University Press (2009), 180-204.
T. Keleti, A $1$-dimensional subset of the reals that intersects each of its translates in at most a single point, Real Anal. Exchange 24 (1998/99) no. 2, 843-844.
T. Keleti, Construction of $1$-dimensional subsets of the reals not containing similar copies of given patterns, Anal. PDE 1 (2008), no. 1, 29-33.
I. Laba, M. Pramanik, Arithmetic progressions in sets of fractional dimension, Geom. Funct. Anal. 19 (2009), 429-456.
P. Mattila, Geometry of sets and measures in Euclidean spaces, Cambridge University Press (1995).
P. Mattila, Integralgeometric properties of capacities, Trans. Amer. Math. Soc. 266 (1981), 539-544.
Mathematical Reviews (MathSciNet): MR617550
P. Mattila, private communication
K. F. Roth: On certain sets of integers, J.
E. Szemerédi, On sets of integers containing no four elements in arithmetic progression, Acta Math. Acad. Sci. Hung. 20 (1969) 89-104.
Mathematical Reviews (MathSciNet): MR245555
E. Szemerédi, On sets of integers containing no k elements in arithmetic progression, Acta Arithmetica 27 (1975), 199-245.
Mathematical Reviews (MathSciNet): MR369312