## Analysis & PDE

• Anal. PDE
• Volume 9, Number 3 (2016), 597-614.

### Finite chains inside thin subsets of $\mathbb{R}^d$

#### Abstract

In a recent paper, Chan, Łaba, and Pramanik investigated geometric configurations inside thin subsets of Euclidean space possessing measures with Fourier decay properties. In this paper we ask which configurations can be found inside thin sets of a given Hausdorff dimension without any additional assumptions on the structure. We prove that if the Hausdorff dimension of $E ⊂ ℝd$, $d ≥ 2$, is greater than $1 2(d + 1)$ then, for each $k ∈ ℤ+$, there exists a nonempty interval $I$ such that, given any sequence ${t1,t2,…,tk : tj ∈ I}$, there exists a sequence of distinct points ${xj}j=1k+1$ such that $xj ∈ E$ and $|xi+1 − xi| = tj$ for $1 ≤ i ≤ k$. In other words, $E$ contains vertices of a chain of arbitrary length with prescribed gaps.

#### Article information

Source
Anal. PDE, Volume 9, Number 3 (2016), 597-614.

Dates
Revised: 23 April 2015
Accepted: 11 October 2015
First available in Project Euclid: 16 November 2017

https://projecteuclid.org/euclid.apde/1510843263

Digital Object Identifier
doi:10.2140/apde.2016.9.597

Mathematical Reviews number (MathSciNet)
MR3518531

Zentralblatt MATH identifier
1342.28006

#### Citation

Bennett, Michael; Iosevich, Alexander; Taylor, Krystal. Finite chains inside thin subsets of $\mathbb{R}^d$. Anal. PDE 9 (2016), no. 3, 597--614. doi:10.2140/apde.2016.9.597. https://projecteuclid.org/euclid.apde/1510843263

#### References

• J. Bourgain, “A Szemerédi type theorem for sets of positive density in ${\R}\sp k$”, Israel J. Math. 54:3 (1986), 307–316.
• V. Chan, I. Łaba, and M. Pramanik, “Finite configurations in sparse sets”, preprint, 2013.
• M. B. Erdõgan, “A bilinear Fourier extension theorem and applications to the distance set problem”, Int. Math. Res. Not. 2005:23 (2005), 1411–1425.
• S. Eswarathasan, A. Iosevich, and K. Taylor, “Fourier integral operators, fractal sets, and the regular value theorem”, Adv. Math. 228:4 (2011), 2385–2402.
• K. J. Falconer, “On the Hausdorff dimensions of distance sets”, Mathematika 32:2 (1985), 206–212.
• K. J. Falconer, The geometry of fractal sets, Cambridge Tracts in Mathematics 85, Cambridge University Press, 1986.
• H. Furstenberg, Y. Katznelson, and B. Weiss, “Ergodic theory and configurations in sets of positive density”, pp. 184–198 in Mathematics of Ramsey theory, edited by J. Nešetřil and V. Rödl, Algorithms and Combinatorics 5, Springer, Berlin, 1990.
• A. Iosevich, M. Mourgoglou, and K. Taylor, “On the Mattila–Sjölin theorem for distance sets”, Ann. Acad. Sci. Fenn. Math. 37:2 (2012), 557–562.
• A. Iosevich, E. Sawyer, K. Taylor, and I. Uriarte-Tuero, “Fractal analogs of classical convolution inequalities”, preprint, 2014.
• P. Maga, “Full dimensional sets without given patterns”, Real Anal. Exchange 36:1 (2010), 79–90.
• P. Mattila, Geometry of sets and measures in Euclidean spaces: fractals and rectifiability, Cambridge Studies in Advanced Mathematics 44, Cambridge University Press, 1995.
• J. Schur, “Bemerkungen zur Theorie der beschränkten Bilinearformen mit unendlich vielen Veränderlichen”, J. Reine Angew. Math. 140 (1911), 1–28.
• E. M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series 43, Princeton University Press, 1993.
• R. S. Strichartz, “Fourier asymptotics of fractal measures”, J. Funct. Anal. 89:1 (1990), 154–187.
• T. H. Wolff, “Decay of circular means of Fourier transforms of measures”, Int. Math. Res. Not. 1999:10 (1999), 547–567.
• T. H. Wolff, Lectures on harmonic analysis, edited by I. Łaba and C. Shubin, University Lecture Series 29, American Mathematical Society, Providence, RI, 2003.
• T. Ziegler, “Nilfactors of $\R\sp m$-actions and configurations in sets of positive upper density in $\R\sp m$”, J. Anal. Math. 99 (2006), 249–266.