Density theorems for anisotropic point configurations. (English) Zbl 1502.05236
Summary: Several results in the existing literature establish Euclidean density theorems of the following strong type. These results claim that every set of positive upper Banach density in the Euclidean space of an appropriate dimension contains isometric copies of all sufficiently large elements of a prescribed family of finite point configurations. So far, all results of this type discussed linear isotropic dilates of a fixed point configuration. In this paper, we initiate the study of analogous density theorems for families of point configurations generated by anisotropic dilations, i.e., families with power-type dependence on a single parameter interpreted as their size. More specifically, we prove nonisotropic power-type generalizations of a result by J. Bourgain [J. Anal. Math. 50, 169–181 (1988; Zbl 0675.42010)] on vertices of a simplex, a result by N. Lyall and Á. Magyar [Anal. PDE 13, No. 3, 685–700 (2020; Zbl 1471.11025)] on vertices of a rectangular box, and a result on distance trees, which is a particular case of the treatise of distance graphs by Lyall and Magyar [loc. cit.]. Another source of motivation for this paper is providing additional evidence for the versatility of the approach stemming from the work of B. Cook et al. [Bull. Lond. Math. Soc. 49, No. 4, 676–689 (2017; Zbl 1370.05210)] and its modification used recently by P. Durcik and V. Kovač [Anal. PDE 15, No. 2, 507–549 (2022; Zbl 1486.05305)]. Finally, yet another purpose of this paper is to single out anisotropic multilinear singular integral operators associated with the above combinatorial problems, as they are interesting on their own.
MSC:
| 05D10 | Ramsey theory |
| 05C55 | Generalized Ramsey theory |
| 05C12 | Distance in graphs |
| 28A75 | Length, area, volume, other geometric measure theory |
| 42B20 | Singular and oscillatory integrals (Calderón-Zygmund, etc.) |
References:
| [1] | Bennett, M., Iosevich, A., and Taylor, K., Finite chains inside thin subsets of ℝ^d. Anal. PDE9(2016), no. 3, 597-614. · Zbl 1342.28006 |
| [2] | Bernicot, F. and Durcik, P., Boundedness of some multi-parameter fiber-wise multiplier operators. Indiana Univ. Math. J. Preprint, 2020. arXiv:2007.02211 |
| [3] | Bourgain, J., A Szemerédi type theorem for sets of positive density inR^k. Israel J. Math.54(1986), no. 3, 307-316. · Zbl 0609.10043 |
| [4] | Bourgain, J., A nonlinear version of Roth’s theorem for sets of positive density in the real line. J. Analyse Math.50(1988), 169-181. · Zbl 0675.42010 |
| [5] | Bulinski, K., Spherical recurrence and locally isometric embeddings of trees into positive density subsets of ℤ^d. Math. Proc. Cambridge Philos. Soc.165(2018), no. 2, 267-278. · Zbl 1407.05063 |
| [6] | Chan, V., Łaba, I., and Pramanik, M., Finite configurations in sparse sets. J. Anal. Math.128(2016), 289-335. · Zbl 1375.28008 |
| [7] | Coifman, R. R. and Meyer, Y., On commutators of singular integrals and bilinear singular integrals. Trans. Amer. Math. Soc.212(1975), 315-331. · Zbl 0324.44005 |
| [8] | Coifman, R. R. and Meyer, Y., Au delà des opérateurs pseudo-différentiels. Astérisque, 57, Société mathématique de France, Paris, 1978. · Zbl 0483.35082 |
| [9] | Coifman, R. R., and Meyer, Y., Commutateurs d’intégrales singulières et opérateurs multilinéaires. Ann. Inst. Fourier (Grenoble)28(1978), no. 3, 177-202. · Zbl 0368.47031 |
| [10] | Cook, B., Magyar, Á., and Pramanik, M., A Roth-type theorem for dense subsets of ℝ^d. Bull. Lon. Math. Soc.49(2017), no. 4, 676-689. · Zbl 1370.05210 |
| [11] | Denson, J., Pramanik, M., and Zahl, J., Large sets avoiding rough patterns. In: M. T. Rassias (ed.), Harmonic Analysis and Applications, Springer Optimization and Its Applications 168, Springer, Cham, 2021, pp. 59-75. · Zbl 07444918 |
| [12] | Durcik, P., An \({\mathsf{L}}^4\) estimate for a singular entangled quadrilinear form. Math. Res. Lett.22(2015), no. 5, 1317-1332. · Zbl 1344.42010 |
| [13] | Durcik, P., L^p estimates for a singular entangled quadrilinear form. Trans. Amer. Math. Soc.369(2017), no. 10, 6935-6951. · Zbl 1377.42015 |
| [14] | Durcik, P., Guo, S., and Roos, J., A polynomial Roth theorem on the real line. Trans. Amer. Math. Soc.371(2019), no. 10, 6973-6993. · Zbl 1432.42005 |
| [15] | Durcik, P. and Kovač, V., Boxes, extended boxes, and sets of positive upper density in the Euclidean space. Math. Proc. Cambridge Philos. Soc. Published Online (2021). doi:10.1017/S0305004120000316 · Zbl 1476.05195 |
| [16] | Durcik, P. and Kovač, V., A Szemerédi-type theorem for subsets of the unit cube. Anal. PDE Preprint, 2020. arXiv:2003.01189 |
| [17] | Durcik, P., Kovač, V., and Rimanić, L., On side lengths of corners in positive density subsets of the Euclidean space. Int. Math. Res. Not.2018(2018), no. 22, 6844-6869. · Zbl 1442.05234 |
| [18] | Durcik, P., Kovač, V., Škreb, K. A., and Thiele, C., Norm-variation of ergodic averages with respect to two commuting transformations. Ergod. Th. & Dynam. Sys.39(2019), no. 3, 658-688. · Zbl 1416.37007 |
| [19] | Durcik, P., Kovač, V., and Thiele, C., Power-type cancellation for the simplex Hilbert transform. J. Anal. Math.139(2019), 67-82. · Zbl 1433.42009 |
| [20] | Durcik, P. and Roos, J., Averages of simplex Hilbert transforms. Proc. Amer. Math. Soc.149(2021), 633-647. · Zbl 1455.42008 |
| [21] | Durcik, P. and Thiele, C., Singular Brascamp-Lieb inequalities with cubical structure. Bull. Lond. Math. Soc.52(2020), no. 2, 283-298. · Zbl 1442.26025 |
| [22] | Durrett, R., Probability—theory and examples. 5th ed., Cambridge Series in Statistical and Probabilistic Mathematics, 49, Cambridge University Press, Cambridge, 2019. · Zbl 1440.60001 |
| [23] | Falconer, K. J. and Marstrand, J. M., Plane sets with positive density at infinity contain all large distances. Bull. Lond. Math. Soc.18(1986), no. 5, 471-474. · Zbl 0599.28008 |
| [24] | Fraser, R., Guo, S., and Pramanik, M., Polynomial Roth theorems on sets of fractional dimensions. Int. Math. Res. Not. Preprint, 2019. arXiv:1904.11123 |
| [25] | Fraser, R. and Pramanik, M., Large sets avoiding patterns. Anal. PDE11(2018), no. 5, 1083-1111. · Zbl 1388.28006 |
| [26] | Furstenberg, H. and Katznelson, Y., An ergodic Szemerédi theorem for commuting transformations. J. Anal. Math.38(1978), no. 1, 275-291. · Zbl 0426.28014 |
| [27] | Furstenberg, H., Katznelson, Y., and Weiss, B., Ergodic theory and configurations in sets of positive density. In: Nešetřil, J. and Rödl, V. (eds.), Mathematics of Ramsey theory, Algorithms and Combinatorics, 5, Springer, Berlin, 1990, pp. 184-198. · Zbl 0738.28013 |
| [28] | Ghosh, A., Bhojak, A., Mohanty, P., and Shrivastava, S., Sharp weighted estimates for multi-linear Calderón-Zygmund operators on non-homogeneous spaces. J. Pseudo-Differ. Oper. Appl.11(2020), 1833-1867. · Zbl 1453.42014 |
| [29] | Grafakos, L. and Torres, R. H., Multilinear Calderón-Zygmund theory. Adv. Math.165(2002), no. 1, 124-164. · Zbl 1032.42020 |
| [30] | Green, B. and Tao, T., The primes contain arbitrarily long arithmetic progressions. Ann. of Math. (2). 167(2008), no. 2, 481-547. · Zbl 1191.11025 |
| [31] | Greenleaf, A., Iosevich, A., Liu, B., and Palsson, E., An elementary approach to simplexes in thin subsets of Euclidean space. Preprint, 2016. arXiv:1608.04777 |
| [32] | Greenleaf, A., Iosevich, A., and Mkrtchyan, S., Existence of similar point configurations in thin subsets of ℝ^d. Math. Z.297(2021), 855-865. · Zbl 1456.52020 |
| [33] | Greenleaf, A., Iosevich, A., and Pramanik, M., On necklaces inside thin subsets of ℝ^d. Math. Res. Lett.24(2017), no. 2, 347-362. · Zbl 1390.28020 |
| [34] | Henriot, K., Łaba, I., and Pramanik, M., On polynomial configurations in fractal sets. Anal. PDE9(2016), no. 5, 1153-1184. · Zbl 1420.28003 |
| [35] | Huckaba, L., Lyall, N., and Magyar, Á., Simplices and sets of positive upper density in ℝ^d. Proc. Amer. Math. Soc.145(2017), no. 6, 2335-2347. · Zbl 1397.11021 |
| [36] | Iosevich, A. and Liu, B., Equilateral triangles in subsets of ℝ^d of large Hausdorff dimension. Israel J. Math.231(2019), no. 1, 123-137. · Zbl 1417.52023 |
| [37] | Iosevich, A. and Magyar, Á., Simplices in thin subsets of Euclidean spaces. Preprint, 2020. arXiv:2009.04902 |
| [38] | Iosevich, A. and Taylor, K., Finite trees inside thin subsets of ℝ^d. In: A. Karapetyants, V. Kravchenko, and E. Liflyand (eds.), Modern methods in operator theory and harmonic analysis, Springer Proceedings in Mathematics & Statistics, 291, Springer, Cham, 2019, pp. 51-56. |
| [39] | Keleti, T., Construction of one-dimensional subsets of the reals not containing similar copies of given patterns. Anal. PDE1(2008), no. 1, 29-33. · Zbl 1151.28302 |
| [40] | Kovač, V., Bellman function technique for multilinear estimates and an application to generalized paraproducts. Indiana Univ. Math. J.60(2011), no. 3, 813-846. · Zbl 1256.42021 |
| [41] | Kovač, V., Boundedness of the twisted paraproduct. Rev. Mat. Iberoam.28(2012), no. 4, 1143-1164. · Zbl 1266.42025 |
| [42] | Kovač, V. and Škreb, K. A., One modification of the martingale transform and its applications to paraproducts and stochastic integrals. J. Math. Anal. Appl.426(2015), no. 2, 1143-1163. · Zbl 1329.60111 |
| [43] | Kovač, V. and Thiele, C., A T(1) theorem for entangled multilinear dyadic Calderón-Zygmund operators. Illinois J. Math.57(2013), no. 3, 775-799. · Zbl 1311.42040 |
| [44] | Łaba, I. and Pramanik, M., Arithmetic progressions in sets of fractional dimension. Geom. Funct. Anal.19(2009), no. 2, 429-456. · Zbl 1184.28010 |
| [45] | Lyall, N. and Magyar, Á., Product of simplices and sets of positive upper density in ℝ^d. Math. Proc. Cambridge Philos. Soc.165(2018), no. 1, 25-51. · Zbl 1411.11009 |
| [46] | Lyall, N. and Magyar, Á., Weak hypergraph regularity and applications to geometric Ramsey theory. Trans. Amer. Math. Soc. Preprint, 2019. |
| [47] | Lyall, N. and Magyar, Á., Distance graphs and sets of positive upper density in ℝ^d. Anal. PDE13(2020), no. 3, 685-700. · Zbl 1471.11025 |
| [48] | Lyall, N. and Magyar, Á., Distances and trees in dense subsets of ℤ^d. Israel J. Math.240(2020), 769-790. · Zbl 1468.11031 |
| [49] | Magyar, Á., k-Point configurations in sets of positive density of ℤ^n. Duke Math. J.146(2009), no. 1, 1-34. · Zbl 1165.05029 |
| [50] | Shkredov, I. D., On a problem of Gowers (in Russian). Izv. Ross. Akad. Nauk Ser. Mat.70(2006), no. 2, 179-221. English translation in Izv. Math. 70(2006), no. 2, 385-425. · Zbl 1149.11006 |
| [51] | Shmerkin, P., Salem sets with no arithmetic progressions. Int. Math. Res. Not. IMRN7(2017), 1929-1941. · Zbl 1405.11011 |
| [52] | Stein, E. M., Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals. Princeton Mathematical Series, 43, Princeton University Press, Princeton, NJ, 1993. · Zbl 0821.42001 |
| [53] | Stein, E. M. and Wainger, S., Problems in harmonic analysis related to curvature. Bull. Amer. Math. Soc.84(1978), no. 6, 1239-1295. · Zbl 0393.42010 |
| [54] | Stein, E. M. and Weiss, G., Introduction to Fourier analysis on Euclidean spaces, Princeton Mathematical Series, 32, Princeton University Press, Princeton, NJ, 1971. · Zbl 0232.42007 |
| [55] | Stipčić, M., T(1) theorem for dyadic singular integral forms associated with hypergraphs. J. Math. Anal. Appl.481(2020), no. 2, Article no. 123496. · Zbl 1423.42033 |
| [56] | Szemerédi, E., On sets of integers containing no k elements in arithmetic progression. Acta Arith.27(1975), 199-245. · Zbl 0303.10056 |
| [57] | Tao, T., The ergodic and combinatorial approaches to Szemerédi’s theorem. In: A. Granville, M. B. Nathanson, and J. Solymosi (eds.), Additive combinatorics, CRM Proceedings and Lecture Notes, 43, American Mathematical Society, Providence, RI, 2007, pp. 145-193. · Zbl 1159.11005 |
| [58] | Tao, T., Cancellation for the multilinear Hilbert transform. Collect. Math.67(2016), no. 2, 191-206. · Zbl 1369.42014 |
| [59] | Yavicoli, A., Patterns in thick compact sets. Israel J. Math. Preprint, 2020. arXiv:1910.10057 |
| [60] | Zorin-Kranich, P., Cancellation for the simplex Hilbert transform. Math. Res. Lett.24(2017), no. 2, 581-592. · Zbl 1387.42014 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
