Dimension of images of large level sets


  • Gavin Armstrong
  • Anthony G. O'Farrell




Let $k$ be a natural number. We consider $k$-times continuously differentiable real-valued functions $f\colon E\to \mathbb{R} $, where $E$ is some interval on the line having positive length. For $0<\alpha <1$ let $I_\alpha (f)$ denote the set of values $y\in \mathbb{R} $ whose preimage $f^{-1}(y)$ has Hausdorff dimension at least $\alpha$. We consider how large can be the Hausdorff dimension of $I_\alpha (f)$, as $f$ ranges over the set of all $k$-times continuously differentiable functions from $E$ into $\mathbb{R} $. We show that the sharp upper bound on $\dim I_\alpha (f)$ is $(1-\alpha )/k$.


Allan, G., Kakiko, G., O'Farrell, A. G., and Watson, R. O., Finitely-generated algebras of smooth functions in one dimension, J. Funct. Anal. 158 (1998), no. 2, 458–74. https://doi.org/10.1006/jfan.1998.3250

Balka, R., Buczolich, Z., and Elekes, M., Topological Hausdorff dimension and fibres of generic continuous functions on fractals, Chaos Solitons Fractals 45 (2012), no. 12, 1579–1589. https://doi.org/10.1016/j.chaos.2012.08.005

Balka, R., Inductive topological Hausdorff dimension and fibers of generic continuous functions, Monatsh. Math. 174 (2014), no. 1, 1–28. https://doi.org/10.1007/s00605-014-0621-7

Balka, R., Darji, U. B., and Elekes, M., Bruckner-Garg-type results with respect to Haar null sets in $C[0,1]$, Proc. Edinb. Math. Soc. (2) 60 (2017), no. 1, 17–30. https://doi.org/10.1017/S0013091515000577

Balka, R., Darji, U. B., and Elekes, M., Hausdorff and packing dimension of fibers and graphs of prevalent continuous maps, Adv. Math. 293 (2016) 221–274. https://doi.org/10.1016/j.aim.2016.02.005

Balka, R., Dimensions of fibres of generic continuous maps Monatsh. Math., 184 (2017), no. 3, 339–378. https://doi.org/10.1007/s00605-017-1067-5

Bertoin, J., Hausdorff dimension of the level sets for self-affine functions, Japan J. Appl. Math. 7 (1990), no. 2, 197–202. https://doi.org/10.1007/BF03167841

Beyer, W. A., Hausdorff dimension of level sets of some Rademacher series, Pacific J. Math. 12 (1962) 35–46 http://projecteuclid.org/euclid.pjm/1103036703

Bishop, C. J., and Peres, Y., Fractals in probability and analysis, Cambridge University Press, Cambridge, 2017. https://doi.org/10.1017/9781316460238

Bruckner, A. M., and Garg, K. M., The level structure of a residual set of continuous functions, Trans. Amer. Math. Soc. 232 (1977) 307–321. https://doi.org/10.2307/1998943

Bruckner, A. M., Differentiation of real functions, Lecture Notes in Mathematics, 659. Springer, Berlin, 1978.

Bruckner, A. M., and Petruska, G., Some typical results on bounded Baire $1$ functions, Acta Math. Hungar. 43 (1984), no. 3–4, 325–333. https://doi.org/10.1007/BF01958029

Buckley, S., Space-filling curves and related functions, Bull. Irish Math. Soc. 36 (1996) 9–18.

Dougherty, R., Examples of non-shy sets, Fund. Math. 144 (1994), no. 1, 73–88. https://doi.org/10.4064/fm-144-1-73-88

Garg, K. M., On a residual set of continuous functions, Czechoslovak Math. J. 20(95) (1970) 537–543.

Garnett, J., and O'Farrell, A. G., Sobolev approximation by a sum of subalgebras on the circle, Pacific J. Math. 65 (1976), no. 1, 55–63. http://projecteuclid.org/euclid.pjm/1102866952

Elekes, M., Level sets of differentiable functions of two variables with non-vanishing gradient, J. Math. Anal. Appl. 270 (2002), no. 2, 369–382. https://doi.org/10.1016/S0022-247X(02)00072-0

Falconer, K., Fractal geometry: Mathematical foundations and applications, Third edition. John Wiley & Sons, Ltd., Chichester, 2014.

Federer, H., Geometric Measure Theory, Die Grundlehren der mathematischen Wissenschaften, Band 153. Springer-Verlag New York Inc., New York 1969.

Howroyd, J. D., On dimension and on the existence of sets of finite positive Hausdorff measure, Proc. London Math. Soc. (3) 70 (1995), no. 3,7 581–604. https://doi.org/10.1112/plms/s3-70.3.581

Hunt, B., The Hausdorff dimension of graphs of Weierstrass functions, Proc. Amer. Math. Soc. 126 (1998), no. 3, 791–800. https://doi.org/10.1090/S0002-9939-98-04387-1

Kirchheim, B., Some further typical results on bounded Baire one functions, Acta Math. Hungar. 62(1993), no. 1–2, 119–129. https://doi.org/10.1007/BF01874223

Kirchheim, B., Typical approximately continuous functions are surprisingly thick, Real Analysis Exchange 18 (1992–93), no. 1, 52–62.

Kirchheim, B., Hausdorff measure and level sets of typical continuous mappings in Euclidean spaces, Trans. Amer. Math. Soc. 347 (1995), no. 5, 1763–1777. https://doi.org/10.2307/2154971

Kôno, N., On self-affine functions, Japan J. Appl. Math. 3 (1986), no. 2, 259–269. https://doi.org/10.1007/BF03167101

Kôno, N., On self-affine functions II, Japan J. Appl. Math. 5 (1988), no. 3, 441–454 https://doi.org/10.1007/BF03167911

Mattila, M., Geometry of sets and measures in Euclidean spaces, Cambridge University Press, Cambridge, 1995. https://doi.org/10.1017/CBO9780511623813

Mauldin, R. D., and Williams, S. C., On the Hausdorff dimension of some graphs, Trans. Amer. Math. Soc. 298 (1986), no. 2, 793–803. https://doi.org/10.2307/2000650

Morse, A. P., The behaviour of a function on its critical set, Ann. of Math. (2) 40 (1939), no. 1, 62–70. https://doi.org/10.2307/1968544

O'Farrell, A. G., $C^infty $ maps can increase $C^infty $ dimension, Invent. Math. 89 (1987), no. 3, 663–668. https://doi.org/10.1007/BF01388989

O'Farrell, A. G., and Roginskaya, M., Conjugacy of real diffeomorphisms. A Survey, Algebra i Analiz, 22 (2010) 3–56 = St. Petersburg Math. J. 22 (2011), no. 1, 1–40. https://doi.org/10.1090/S1061-0022-2010-01130-0

Orponen, T., On the packing measure of slices of self-similar sets, J. Fractal Geom. 2 (2013), no. 4, 389–401. https://doi.org/10.4171/JFG/26

Rogers, C. A., Hausdorff measures, Cambridge University Press, London-New York 1970.

Shen, W., Hausdorff dimension of the graphs of the classical Weierstrass functions, Math. Z. 289 (2018), no-1 2, 223–266. https://doi.org/10.1007/s00209-017-1949-1



How to Cite

Armstrong, G., & O’Farrell, A. G. . (2022). Dimension of images of large level sets. MATHEMATICA SCANDINAVICA, 128(1). https://doi.org/10.7146/math.scand.a-129246