Abstract
We prove new results of Mattila–Sjölin type, giving lower bounds on Hausdorff dimensions of thin sets \(E\subset \mathbb {R}^d\) ensuring that various k-point configuration sets, generated by elements of E, have nonempty interior. The dimensional thresholds in our previous work (Greenleaf et al., Mathematika 68(1):163–190, 2022) were dictated by associating to a configuration function a family of generalized Radon transforms, and then optimizing \(L^2\)-Sobolev estimates for them over all nontrivial bipartite partitions of the k points. In the current work, we extend this by allowing the optimization to be done locally over the configuration’s incidence relation, or even microlocally over the conormal bundle of the incidence relation. We use this approach to prove Mattila–Sjölin type results for (i) areas of subtriangles determined by quadrilaterals and pentagons in a set \(E\subset \mathbb {R}^2\); (ii) pairs of ratios of distances of 4-tuples in \(\mathbb {R}^d\); and (iii) similarity classes of triangles in \(\mathbb {R}^d\), as well as to (iv) give a short proof of Palsson and Romero Acosta’s result on congruence classes of triangles in \(\mathbb {R}^d\).
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References
Bennett, M., Iosevich, A., Taylor, K.: Finite chains inside thin subsets of \({{\mathbb{R} }}^d\). Anal. PDE 9(3), 597–614 (2016)
Borges, T., Iosevich, A., Ou, Y.: A singular variant of the Falconer distance problem (2023). arXiv:2306.05247
Chan, V., Łaba, I., Pramanik, M.: Finite configurations in sparse sets. J. d’Analyse Math. 128, 289–335 (2016)
Cheong, D., Koh, D., Pham, T., Shen, C.Y.: Mattila–Sjölin type functions: a finite field model. Vietnam J. Math. (2021). https://doi.org/10.1007/s10013-021-00538-z
Falconer, K.J.: On the Hausdorff dimensions of distance sets. Mathematika 32, 206–212 (1986)
Gaitan, J., Greenleaf, A., Palsson, E., Psaromiligkos, G.: On restricted Falconer distance sets. Canad. J. Math. (2024) (to appear). arXiv:2305.18053v2
Grafakos, L., Greenleaf, A., Iosevich, A., Palsson, E.: Multilinear generalized Radon transforms and point configurations. Forum Math. 27(4), 2323–2360 (2015)
Greenleaf, A., Iosevich, A., Pramanik, M.: On necklaces inside thin subsets of \({\mathbb{R} }^d\). Math. Res. Lett. 24(2), 347–362 (2017)
Greenleaf, A., Iosevich, A., Taylor, K.: Configuration sets with nonempty interior. J. Geom. Anal. 31(7), 6662–6680 (2021). https://doi.org/10.1007/s12220-019-00288-y
Greenleaf, A., Iosevich, A., Taylor, K.: On \(k\)-point configuration sets with nonempty interior. Mathematika 68(1), 163–190 (2022). https://doi.org/10.1112/mtk.12114
Greenleaf, A., Seeger, A.: Fourier integral operators with fold singularities. J. Reine U. Angew. Math. 455, 35–56 (1994)
Guillemin, V., Sternberg, S.: Geometric asymptotics. Math. Surveys 14. Amer. Math. Soc., Providence, R.I. (1977)
Guillemin, V., Sternberg, S.: Some problems in integral geometry and some related problems in microlocal analysis. Am. J. Math. 101(4), 915–955 (1979)
Helgason, S.: The Radon transform on Euclidean spaces, compact two-point homogeneous spaces and Grassmann manifolds. Acta Math. 113, 153–180 (1965)
Hörmander, L.: Fourier integral operators. I. Acta Math. 127(1–2), 79–183 (1971)
Hörmander, L.: The analysis of linear partial differential operators, III and IV. Grund. math. Wissen. 274 and 275. Springer, Berlin (1985)
Iosevich, A., Liu, B.: Pinned distance problem, slicing measures, and local smoothing estimates. Trans. Am. Math. Soc. 371(6), 4459–4474 (2019)
Iosevich, A., Mourgoglou, M., Taylor, K.: On the Mattila-Sjölin theorem for distance sets. Ann. Acad. Sci. Fenn. Math. 37(2), 557–562 (2012)
Iosevich, A., Taylor, K.: Finite trees inside thin subsets of \({\mathbb{R} }^d\), Modern Methods in Operator Theory and Harmonic Analysis, Springer Proc. Math. Stat. 291, 51–56 (2019)
Iosevich, A., Taylor, K., Uriarte-Tuero, I.: Pinned geometric configurations in Euclidean space and Riemannian manifolds. Mathematics 9, 1802 (2021)
Koh, D., Pham, T., Shen, C.-Y.: On the Mattila–Sjölin Distance Theorem for Product Sets. arXiv:2103.11418 (2021)
Mattila, P.: Geometry of sets and measures in Euclidean spaces. Fractals and rectifiability. Cambridge Studies in Adv. Math. 44. Cambridge Univ. Press, Cambridge (1995)
Mattila, P.: Fourier analysis and Hausdorff dimension. Cambridge Studies in Adv. Math. 150. Cambridge Univ. Press (2015)
Mattila, P., Sjölin, P.: Regularity of distance measures and sets. Math. Nachr. 204, 157–162 (1999)
McDonald, A.: Areas spanned by point configurations in the plane. Proc. Am. Math. Soc. 149(5), 2035–2049 (2021)
McDonald, A., Taylor, K.: Finite point configurations in products of thick Cantor sets and a robust nonlinear Newhouse gap lemma. Math. Proc. Cambr. Philos. Soc. 175, 285–301 (2023)
McDonald, A., Taylor, K.: Constant gap length trees in products of thick Cantor sets. In: Proc. Royal Soc. Edinburgh, Section A: Math, to appear (2022). arXiv:2211.10750
Palsson, E., Acosta, F. Romero: A Mattila–Sjölin theorem for simplices in low dimensions (2022). arXiv:2208.07198
Palsson, E., Acosta, F. Romero: A Mattila–Sjölin theorem for triangles. J. Funct. Anal. 284(6), 20 (2023) (Paper No. 109814)
Peres, Y., Schlag, W.: Smoothness of projections, Bernoulli convolutions, and the dimension of exceptions. Duke Math. J. 102(2), 193–251 (2000)
Simon, K., Taylor, K.: Interior of sums of planar sets and curves. Math. Proc. Cambr. Phil. Soc. 168(1), 119–148 (2020)
Steinhaus, H.: Sur les distances des points dans les ensembles de mesure positive. Fund. Math. 1, 93–104 (1920)
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AG is supported in part by US National Science Foundation DMS-1906186 and -2204943, AI by NSF HDR TRIPODS-1934962 and DMS-2154232, and KT by Simons Foundation Grant 523555. The authors thank the referee for helpful suggestions and questions which improved the paper.
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Greenleaf, A., Iosevich, A. & Taylor, K. Nonempty interior of configuration sets via microlocal partition optimization. Math. Z. 306, 66 (2024). https://doi.org/10.1007/s00209-024-03466-z
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DOI: https://doi.org/10.1007/s00209-024-03466-z