Hausdorff dimension of Cantor sets and polynomial hulls

Hausdorff dimension of Cantor sets and polynomial hulls
HTML articles powered by AMS MathViewer

by Burglind Jöricke PDF

Proc. Amer. Math. Soc. 134 (2006), 1347-1354 Request permission

We give examples of Cantor sets in $\mathbb {C}^n$ of Hausdorff dimension 1 whose polynomial hulls have non-empty interior.

References

H. Alexander, Polynomial approximation and hulls in sets of finite linear measure in Cn, Amer. J. Math. 93 (1971), 65–74. MR 284617, DOI 10.2307/2373448
H. Alexander, The polynomial hull of a set of finite linear measure in $\textbf {C}^n$, J. Analyse Math. 47 (1986), 238–242. MR 874052, DOI 10.1007/BF02792540
Tien-Cuong Dinh and Mark G. Lawrence, Polynomial hulls and positive currents, Ann. Fac. Sci. Toulouse Math. (6) 12 (2003), no. 3, 317–334 (English, with English and French summaries). MR 2030090, DOI 10.5802/afst.1051
Henkin G.M., Oral communication.
Burglind Jöricke, A Cantor set in the unit sphere in $\Bbb C^2$ with large polynomial hull, Michigan Math. J. 53 (2005), no. 1, 189–207. MR 2125541, DOI 10.1307/mmj/1114021092
Rudin W., Function algebras, (Proc. Internat. Sympos., Tulane Univ., 1965), Scott, Foresman, Chicago, IL, 1966.
Nessim Sibony, Sur la frontière de Shilov des domaines de $\textbf {C}^n$, Math. Ann. 273 (1985), no. 1, 115–121 (French, with English summary). MR 814198, DOI 10.1007/BF01455917
Gabriel Stolzenberg, Uniform approximation on smooth curves, Acta Math. 115 (1966), 185–198. MR 192080, DOI 10.1007/BF02392207
A. G. Vituškin, A certain problem of W. Rudin, Dokl. Akad. Nauk SSSR 213 (1973), 14–15 (Russian). MR 0333243
John Wermer, The hull of a curve in $C^{n}$, Ann. of Math. (2) 68 (1958), 550–561. MR 100102, DOI 10.2307/1970155