## On Falconer’s distance set conjecture.(English)Zbl 1141.42007

Let $$E$$ be a compact subset of $$\mathbb{R}^d$$ and $$\Delta(E)=\{|x-y| : x,y \in E\}$$. Falconer’s conjecture is that if the Hausdorff dimension of $$E$$ is greater than $$d/2$$, then the Lebesgue measure of $$\Delta(E)$$ is positive. In this paper the author proves that this is true if $$d > 2$$ and $$\dim(E) > d(d+2)/2(d+1)$$.

### MSC:

 42B10 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
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### References:

 [1] Aronov, B., Pach, J., Sharir, M. and Tardos, G.: Distinct distances in three and higher dimensions. In Proceedings of the 35th Annual ACM Symposium on Theory of Computing , 541-546. ACM, New York, 2003. · Zbl 1192.52024 [2] Arutyunyants, G. and Iosevich, A.: Falconer conjecture, spherical averages and discrete analogs. In Towards a theory of geometric graphs , 15-24. Contemp. Math. 342 . Amer. Math. Soc., Providence, 2004. · Zbl 1096.28003 [3] Bourgain, J.: Hausdorff dimension and distance sets. Israel J. Math. 87 (1994), 193-201. · Zbl 0807.28004 [4] Bourgain, J.: On the Erdös-Volkmann and Katz-Tao ring conjectures. Geom. Funct. Anal. 13 (2003), 334-365. · Zbl 1115.11049 [5] Carleson, L. and Sjölin, P.: Oscillatory integrals and a multiplier problem for the disc. Studia Math. 44 (1972), 287-299. · Zbl 0215.18303 [6] Erdo\~gan, M. B.: A note on the Fourier transform of fractal measures. Math. Res. Lett. 11 (2004), no. 2-3, 299-313. · Zbl 1083.42006 [7] Erdös, P.: On sets of distances of $$n$$ points. Amer. Math. Monthly 53 (1946), 248-250. JSTOR: · Zbl 0060.34805 [8] Falconer, K. J.: On the Hausdorff dimension of distance sets. Mathematika 32 (1985), 206-212. · Zbl 0605.28005 [9] Hofmann, S. and Iosevich, A.: Circular averages and Falconer/Erdös distance conjecture in the plane for random metrics. Proc. Amer. Math. Soc. 133 (2005), no. 1, 133-143. · Zbl 1096.28004 [10] Iosevich, A. and Laba, I.: $$K$$-distance sets, Falconer conjecture and discrete analogs. Integers 5 (2005), no. 2, A8, 11 pp. (electronic). · Zbl 1139.28002 [11] Kahane, J. P.: Some random series of functions . Second edition. Cambridge Studies in Advanced Mathematics 5 . Cambridge Univ. Press, 1985. · Zbl 0571.60002 [12] Katz, N. H. and Tao, T.: Some connections between Falconer’s distance set conjecture and sets of Furstenburg type. New York J. Math. 7 (2001), 149-187. · Zbl 0991.28006 [13] Katz, N. H. and Tardos, G.: A new entropy inequality for the Erdös distance problem. In Towards a theory of geometric graphs , 119-126. Contemp. Math. 342 . Amer. Math. Soc., Providence, RI, 2004. · Zbl 1069.52017 [14] Mattila, P.: Spherical averages of Fourier transforms of measures with finite energy; dimension of intersections and distance sets. Mathematika 34 (1987), 207-228. · Zbl 0645.28004 [15] Mattila, P.: Geometry of sets and measures in Euclidean spaces. Fractals and rectifiability . Cambridge Studies in Advanced Mathematics 44 . Cambridge University Press, Cambridge, 1995. · Zbl 0819.28004 [16] Mattila, P. and Sjölin, P.: Regularity of distance measures and sets. Math. Nachr. 204 (1999), 157-162. · Zbl 1050.42009 [17] Pach, J. and Agarwal, P. K.: Combinatorial geometry . Wiley Interscience Series in Discrete Mathematics and Optimization. John Wiley, New York, 1995. [18] Peres, Y. and Schlag, W.: Smoothness of projections, Bernoulli convolutions, and the dimension of exceptions. Duke Math. J. 102 (2000), no. 2, 193-251. · Zbl 0961.42007 [19] Salem, R.: Algebraic numbers and Fourier analysis . D. C. Heath and Co., Boston, Mass., 1963. · Zbl 0126.07802 [20] Sjölin, P.: Estimates of spherical averages of Fourier transforms and dimensions of sets. Mathematika 40 (1993), 322-330. · Zbl 0789.28006 [21] Sjölin, P. and Soria, F.: Estimates of averages of Fourier transforms with respect to general measures. Proc. Roy. Soc. Edinburg Sect. A 133 (2003), 943-950. · Zbl 1051.42011 [22] Tao, T.: A sharp bilinear restrictions estimate for paraboloids. Geom. Funct. Anal. 13 (2003), 1359-1384. · Zbl 1068.42011 [23] Tao, T., Vargas, A. and Vega, L.: A bilinear approach to the restriction and Kakeya conjectures. J. Amer. Math. Soc. 11 (1998), 967-1000. JSTOR: · Zbl 0924.42008 [24] Wolff, T.: Decay of circular means of Fourier transforms of measures. Internat. Math. Res. Notices , 1999, 547-567. · Zbl 0930.42006 [25] Wolff, T.: A sharp bilinear cone restriction estimate. Ann. of Math. (2) 153 (2001), 661-698. JSTOR: · Zbl 1125.42302 [26] Wolff, T.: Lectures on harmonic analysis . University Lecture Series 29 . American Mathematical Society, Providence, RI, 2003.
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