The Denjoy-Clarkson property with respect to Hausdorff measures for the gradient mapping of functions of several variables
Annales de l'Institut Fourier, Volume 58 (2008) no. 2, p. 405-428

We construct a differentiable function f:R n R (n2) such that the set (f) -1 (B(0,1)) is a nonempty set of Hausdorff dimension 1. This answers a question posed by Z. Buczolich.

On construit une fonction différentiable f:R n R (n2) telle que l’ensemble (f) -1 (B(0,1)) est non vide et sa dimension de Hausdorff est 1. C’est une réponse à une question posée par Z. Buczolich.

DOI : https://doi.org/10.5802/aif.2356
Classification:  26B05,  28A75
Keywords: Denjoy–Clarkson property, gradient, Hausdorff measure, infinite game
@article{AIF_2008__58_2_405_0,
     author = {Zelen\'y, Miroslav},
     title = {The Denjoy-Clarkson property with respect to Hausdorff measures for the gradient mapping of functions of several variables},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {58},
     number = {2},
     year = {2008},
     pages = {405-428},
     doi = {10.5802/aif.2356},
     mrnumber = {2410378},
     zbl = {1154.26016},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2008__58_2_405_0}
}
Zelený, Miroslav. The Denjoy-Clarkson property with respect to Hausdorff measures for the gradient mapping of functions of several variables. Annales de l'Institut Fourier, Volume 58 (2008) no. 2, pp. 405-428. doi : 10.5802/aif.2356. http://www.numdam.org/item/AIF_2008__58_2_405_0/

[1] Bruckner, A. Differentiation of real functions, American Mathematical Society, Providence, RI, CRM Monograph Series, Tome 5 (1994) | MR 1274044 | Zbl 0796.26004

[2] Buczolich, Z. The n-dimensional gradient has the 1-dimensional Denjoy-Clarkson property, Real Anal. Exchange, Tome 18 (1992-93), pp. 221-224 | MR 1205514 | Zbl 0783.26010

[3] Buczolich, Z. Another note on the gradient problem of C. E. Weil, Real Anal. Exchange, Tome 22 (1996-97), pp. 775-784 | MR 1460988 | Zbl 0940.26011

[4] Buczolich, Z. Solution to the gradient problem of C. E. Weil, Rev. Mat. Iberoamericana, Tome 21 (2005), pp. 889-910 | MR 2231014 | Zbl 1116.26007 | Zbl 05034353

[5] Clarkson, J. A. A property of derivatives, Bull. Amer. Math. Soc., Tome 53 (1947), p. 124-125 | Article | MR 19712 | Zbl 0032.27102

[6] Denjoy, A. Sur une proprieté des fonctions dérivées, Enseignement Math., Tome 18 (1916), pp. 320-328 | JFM 46.0381.05

[7] Deville, R.; Matheron, É. Infinite games, Banach space geometry and the eikonal equation, Proc. Lond. Math. Soc. (3), Tome 95 (2007) no. 1, pp. 49-68 | Article | MR 2329548 | Zbl 1163.91007

[8] Engelking, R. General Topology, Heldermann Verlag, Berlin (1989) | MR 1039321 | Zbl 0684.54001

[9] Holický, P.; Malý, J.; Weil, C. E.; Zajíček, L. A note on the gradient problem, Real Anal. Exchange, Tome 22 (1996-97), pp. 225-235 | MR 1433610 | Zbl 0879.26041

[10] Malý, J. The Darboux property for gradients, Real Anal. Exchange, Tome 22 (1996/97) no. 1, pp. 167-173 | MR 1433604 | Zbl 0879.26042

[11] Malý, J.; Zelený, M. A note on Buczolich’s solution of the Weil gradient problem: a construction based on an infinite game, Acta Math. Hungar., Tome 113 (2006), pp. 145-158 | Article | Zbl 05150231

[12] Mattila, P. Geometry of sets and measures in Euclidean spaces, Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, Tome 44 (1995) | MR 1333890 | Zbl 0819.28004

[13] Weil, C. E. Query 1, Real Anal. Exchange, Tome 16 (1990-91), pp. 373