Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-25T06:57:41.803Z Has data issue: false hasContentIssue false
  • English
  • Français

Density theorems for anisotropic point configurations

Part of: Classical measure theory Extremal combinatorics Harmonic analysis in several variables

Published online by Cambridge University Press:  03 May 2021

Vjekoslav Kovač
Vjekoslav Kovač*
Affiliation:
Department of Mathematics, Faculty of Science, University of Zagreb, Bijenička cesta 30, 10000Zagreb, Croatia
*
e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

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 Bourgain on vertices of a simplex, a result by Lyall and Magyar 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. Another source of motivation for this paper is providing additional evidence for the versatility of the approach stemming from the work of Cook, Magyar, and Pramanik and its modification used recently by Durcik and the present author. 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.

Keywords

Euclidean Ramsey theorypoint configurationdistance graphheat flowsingular integral

MSC classification

Primary: 05D10: Ramsey theory
Secondary: 28A75: Length, area, volume, other geometric measure theory 42B20: Singular and oscillatory integrals (Calderón-Zygmund, etc.)
Type
Article
Information
Canadian Journal of Mathematics , Volume 74 , Issue 5 , October 2022 , pp. 1244 - 1276
Check for updates
Copyright
© Canadian Mathematical Society 2021

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

This work is supported in part by the Croatian Science Foundation project UIP-2017-05-4129 (MUNHANAP) and in part by the Fulbright Scholar Program.

References

Bennett, M., Iosevich, A., and Taylor, K., Finite chains inside thin subsets of ℝ d . Anal. PDE 9(2016), no. 3, 597–614.CrossRefGoogle Scholar
Bernicot, F. and Durcik, P., Boundedness of some multi-parameter fiber-wise multiplier operators. Indiana Univ. Math. J. Preprint, 2020. arXiv:2007.02211 Google Scholar
Bourgain, J., A Szemerédi type theorem for sets of positive density in R k . Israel J. Math. 54(1986), no. 3, 307–316.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
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.Google Scholar
Chan, V., Łaba, I., and Pramanik, M., Finite configurations in sparse sets. J. Anal. Math. 128(2016), 289–335.CrossRefGoogle Scholar
Coifman, R. R. and Meyer, Y., On commutators of singular integrals and bilinear singular integrals. Trans. Amer. Math. Soc. 212(1975), 315–331.CrossRefGoogle Scholar
Coifman, R. R. and Meyer, Y., Au delà des opérateurs pseudo-différentiels. Astérisque, 57, Société mathématique de France, Paris, 1978.Google Scholar
Coifman, R. R., and Meyer, Y., Commutateurs d’intégrales singulières et opérateurs multilinéaires. Ann. Inst. Fourier (Grenoble) 28(1978), no. 3, 177–202.CrossRefGoogle Scholar
Cook, B., Magyar, Á., and Pramanik, M., A Roth-type theorem for dense subsets of ℝd . Bull. Lon. Math. Soc. 49(2017), no. 4, 676–689.CrossRefGoogle Scholar
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.Google Scholar
Durcik, P., An ${\mathsf{L}}^4$ estimate for a singular entangled quadrilinear form . Math. Res. Lett. 22(2015), no. 5, 1317–1332.CrossRefGoogle Scholar
Durcik, P., L p estimates for a singular entangled quadrilinear form . Trans. Amer. Math. Soc. 369(2017), no. 10, 6935–6951.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
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 Google Scholar
Durcik, P. and Kovač, V., A Szemerédi-type theorem for subsets of the unit cube. Anal. PDE Preprint, 2020. arXiv:2003.01189 Google Scholar
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.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
Durcik, P., Kovač, V., and Thiele, C., Power-type cancellation for the simplex Hilbert transform. J. Anal. Math. 139(2019), 67–82.CrossRefGoogle Scholar
Durcik, P. and Roos, J., Averages of simplex Hilbert transforms. Proc. Amer. Math. Soc. 149(2021), 633–647.CrossRefGoogle Scholar
Durcik, P. and Thiele, C., Singular Brascamp–Lieb inequalities with cubical structure. Bull. Lond. Math. Soc. 52(2020), no. 2, 283–298.Google Scholar
Durrett, R., Probability—theory and examples. 5th ed., Cambridge Series in Statistical and Probabilistic Mathematics, 49, Cambridge University Press, Cambridge, 2019.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
Fraser, R., Guo, S., and Pramanik, M., Polynomial Roth theorems on sets of fractional dimensions. Int. Math. Res. Not. Preprint, 2019. arXiv:1904.11123 Google Scholar
Fraser, R. and Pramanik, M., Large sets avoiding patterns. Anal. PDE 11(2018), no. 5, 1083–1111.CrossRefGoogle Scholar
Furstenberg, H. and Katznelson, Y., An ergodic Szemerédi theorem for commuting transformations. J. Anal. Math. 38(1978), no. 1, 275–291.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
Grafakos, L. and Torres, R. H., Multilinear Calderón–Zygmund theory. Adv. Math. 165(2002), no. 1, 124–164.CrossRefGoogle Scholar
Green, B. and Tao, T., The primes contain arbitrarily long arithmetic progressions. Ann. of Math. (2). 167(2008), no. 2, 481–547.CrossRefGoogle Scholar
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 CrossRefGoogle Scholar
Greenleaf, A., Iosevich, A., and Mkrtchyan, S., Existence of similar point configurations in thin subsets of ℝ d . Math. Z. 297(2021), 855–865.Google Scholar
Greenleaf, A., Iosevich, A., and Pramanik, M., On necklaces inside thin subsets of ℝ d . Math. Res. Lett. 24(2017), no. 2, 347–362.CrossRefGoogle Scholar
Henriot, K., Łaba, I., and Pramanik, M., On polynomial configurations in fractal sets. Anal. PDE 9(2016), no. 5, 1153–1184.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
Iosevich, A. and Magyar, Á., Simplices in thin subsets of Euclidean spaces. Preprint, 2020. arXiv:2009.04902 Google Scholar
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.CrossRefGoogle Scholar
Keleti, T., Construction of one-dimensional subsets of the reals not containing similar copies of given patterns. Anal. PDE 1(2008), no. 1, 29–33.CrossRefGoogle Scholar
Kovač, V., Bellman function technique for multilinear estimates and an application to generalized paraproducts. Indiana Univ. Math. J. 60(2011), no. 3, 813–846.CrossRefGoogle Scholar
Kovač, V., Boundedness of the twisted paraproduct. Rev. Mat. Iberoam. 28(2012), no. 4, 1143–1164.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
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.Google Scholar
Łaba, I. and Pramanik, M., Arithmetic progressions in sets of fractional dimension. Geom. Funct. Anal. 19(2009), no. 2, 429–456.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
Lyall, N. and Magyar, Á., Weak hypergraph regularity and applications to geometric Ramsey theory. Trans. Amer. Math. Soc. Preprint, 2019.Google Scholar
Lyall, N. and Magyar, Á., Distance graphs and sets of positive upper density in ℝ d . Anal. PDE 13(2020), no. 3, 685–700.Google Scholar
Lyall, N. and Magyar, Á., Distances and trees in dense subsets of ℤ d . Israel J. Math. 240(2020), 769–790.CrossRefGoogle Scholar
Magyar, Á., k-Point configurations in sets of positive density of ℤ n . Duke Math. J. 146(2009), no. 1, 1–34.Google Scholar
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.Google Scholar
Shmerkin, P., Salem sets with no arithmetic progressions. Int. Math. Res. Not. IMRN 7(2017), 1929–1941.Google Scholar
Stein, E. M., Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals. Princeton Mathematical Series, 43, Princeton University Press, Princeton, NJ, 1993.Google Scholar
Stein, E. M. and Wainger, S., Problems in harmonic analysis related to curvature. Bull. Amer. Math. Soc. 84(1978), no. 6, 1239–1295.CrossRefGoogle Scholar
Stein, E. M. and Weiss, G., Introduction to Fourier analysis on Euclidean spaces, Princeton Mathematical Series, 32, Princeton University Press, Princeton, NJ, 1971.Google Scholar
Stipčić, M., T(1) theorem for dyadic singular integral forms associated with hypergraphs . J. Math. Anal. Appl. 481(2020), no. 2, Article no. 123496.CrossRefGoogle Scholar
Szemerédi, E., On sets of integers containing no k elements in arithmetic progression. Acta Arith. 27(1975), 199–245.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
Tao, T., Cancellation for the multilinear Hilbert transform. Collect. Math. 67(2016), no. 2, 191–206.CrossRefGoogle Scholar
Yavicoli, A., Patterns in thick compact sets. Israel J. Math. Preprint, 2020. arXiv:1910.10057 CrossRefGoogle Scholar
Zorin-Kranich, P., Cancellation for the simplex Hilbert transform. Math. Res. Lett. 24(2017), no. 2, 581–592.CrossRefGoogle Scholar