Finite point configurations in products of thick Cantor sets and a robust nonlinear Newhouse Gap Lemma. (English) Zbl 07725804
Summary: In this paper we prove that the set \(\{|x^1-x^2|,\dots,|x^k-x^{k+1}|:x^i\in E\}\) has non-empty interior in \(\mathbb{R}^k\) when \(E\subset\mathbb{R}^2\) is a Cartesian product of thick Cantor sets \(K_1,K_2\subset\mathbb{R}\). We also prove more general results where the distance map \(|x-y|\) is replaced by a function \(\phi(x,y)\) satisfying mild assumptions on its partial derivatives. In the process, we establish a nonlinear version of the classic Newhouse Gap Lemma, and show that if \(K_1,K_2, \phi\) are as above then there exists an open set \(S\) so that \(\bigcap_{x\in S}\phi(x,K_1\times K_2)\) has non-empty interior.
MSC:
| 28A75 | Length, area, volume, other geometric measure theory |
| 28A80 | Fractals |
| 42Bxx | Harmonic analysis in several variables |
References:
| [1] | Bennett, M., Iosevich, A. and Taylor, K.. Finite chains inside thin subsets of \(\mathbb{R}^d \) . Anal. PDE9(3) 2016, 597-614. · Zbl 1342.28006 |
| [2] | Bourgain, J.. A Szemerédi type theorem for sets of positive density in \(\textbf{R}^k \) . Israel J. Math., 54(3) 1986, 307-316. · Zbl 0609.10043 |
| [3] | Broderick, R., Fishman, L. and Simmons, D.. Quantitative results using variants of Schmidt’s game: dimension bounds, arithmetic progressions, and more. Acta Arith., 188(3) 2019, 289-316. · Zbl 1443.11144 |
| [4] | Chan, V., Łaba, I. and Pramanik, M.. Finite configurations in sparse sets. J. Anal. Math., 1282016, 289-335. · Zbl 1375.28008 |
| [5] | Eswarathasan, S., Iosevich, A. and Taylor, K.. Fourier integral operators, fractal sets, and the regular value theorem. Adv. Math., 228(4) 2011, 2385-2402. · Zbl 1239.28004 |
| [6] | Falconer, K. and Yavicoli, A.. Intersections of thick compact sets in \(\mathbb{R}^d \) . Math. Z., 301(3) 20222291-2315. · Zbl 1514.11017 |
| [7] | Falconer, K. J.. On the Hausdorff dimensions of distance sets. Mathematika, 32(2) (1986), 1985, 206-212. · Zbl 0605.28005 |
| [8] | Furstenberg, H., Katznelson, Y. and Weiss, B.. Ergodic theory and configurations in sets of positive density. In Mathematics of Ramsey theory Algorithms Combin. (Springer, Berlin, 1990), 184-198. · Zbl 0738.28013 |
| [9] | Galo, B. and Mcdonald, A.. Volumes spanned by k-point configurations in \(\mathbb{R}^d \) . arXiv:2102.02323, 2021. · Zbl 1528.28008 |
| [10] | Grafakos, L., Greenleaf, A., Iosevich, A. and Palsson, E.. Multilinear generalised Radon transforms and point configurations. Forum Math., 27(4) 2015, 2323-2360. · Zbl 1317.44002 |
| [11] | Greenleaf, A., Iosevich, A. and Pramanik, M.. On necklaces inside thin subsets of \(\mathbb{R}^d \) . Math. Res. Lett., 24(2) 2017, 347-362. · Zbl 1390.28020 |
| [12] | Greenleaf, A., Iosevich, A. and Taylor, K.. Configuration sets with nonempty interior. J. Geom. Anal., 31(7) 2021, 6662-6680. · Zbl 1472.35470 |
| [13] | Greenleaf, A., Iosevich, A. and Taylor, K.. On k-point configuration sets with nonempty interior. Mathematika, 68(1) 2022, 163-190 · Zbl 1472.35470 |
| [14] | Guth, L., Iosevich, A., Ou, Y. and Wang, H.. On Falconer’s distance set problem in the plane. Invent. Math., 219(3) 2020, 779-830. · Zbl 1430.28001 |
| [15] | Iosevich, A. and Liu, B.. Equilateral triangles in subsets of \(\mathbb{R}^d\) of large Hausdorff dimension. Israel J. Math., 231(1) 2019, 123-137. · Zbl 1417.52023 |
| [16] | Iosevich, A. and Magyar, A.. Simplices in thin subsets of Euclidean spaces. arXiv:2009.04902. |
| [17] | Iosevich, A. and Taylor, K.. Finite trees inside thin subsets of \(\mathbb{R}^d \) . In Modern methods in operator theory and harmonic analysis. Springer Proc. Math. Stat., vol. 291 (Springer, Cham, 2019), 51-56. |
| [18] | Mcdonald, A.. Areas spanned by point configurations in the plane. Proc. Amer. Math. Soc., 149(5) 2021, 2035-2049. · Zbl 1460.52018 |
| [19] | Mcdonald, A. and Taylor, K.. Constant gap length trees in products of thick cantor sets. arXiv:2211.10750. · Zbl 07725804 |
| [20] | Newhouse, S. E.. Nondensity of axiom \(\text{A} \) (a) on \(S^2\) . In Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968) (Amer. Math. Soc., Providence, R.I., 1970). |
| [21] | Ou, Y. and Taylor, K.. Finite point configurations and the regular value theorem in a fractal setting. arXiv:2003.06218. To appear in Indiana J. Math. · Zbl 1527.42010 |
| [22] | Palis, J. and Takens, F.. Hyperbolicity and sensitive chaotic dynamics at homoclinic bifurcations. (Cambridge Studies in Advanced Math. vol 35 (Cambridge University Press, Cambridge, 1993). Fractal dimensions and infinitely many attractors. · Zbl 0790.58014 |
| [23] | Simon, K. and Taylor, K.. Interior of sums of planar sets and curves. Math. Proc. Camb. Phil. Soc., 168(1) 2020, 119-148. · Zbl 1429.28017 |
| [24] | Steinhaus, H.. Sur les distances des points dans les ensembles de mesure positive. Fundamenta Mathematicae, 1(1) 1920, 93-104. · JFM 47.0179.02 |
| [25] | Yavicoli, A.. Patterns in thick compact sets. Israel J. Math., 244(1) 2021, 95-126. · Zbl 1483.28012 |
| [26] | Yavicoli, A.. A survey on newhouse thickness, fractal intersections and patterns. arXiv:2212.02023. |
| [27] | Yavicoli, A.. Thickness and a gap lemma in \(\mathbb{R}^d \) . arXiv:2204.08428. · Zbl 1483.28012 |
| [28] | Ziegler, T.. Nilfactors of \(\mathbb{R}^m \) -actions and configurations in sets of positive upper density in \(\mathbb{R}^m \) . J. Anal. Math., 992006, 249-266. · Zbl 1145.37005 |
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