We study the Riesz -capacity of the so called Dobiński set. We characterize the values of the parameters and for which the -Riesz capacity of the Dobiński set is positive. In particular we show that the Dobiński set has positive logarithmic capacity, thus answering a question of Dayan, Fernandéz and González. We approach the problem by considering the dyadic analogues of the Riesz -capacities which seem to be better adapted to the problem.
Révisé le :
Accepté le :
Publié le :
DOI : 10.5802/crmath.332
Mots clés : Riesz capacity, Logarithmic capacity, Dobiński set, Dyadic capacity, Non-linear capacity, Diophantine approxmation
@article{CRMATH_2022__360_G6_679_0,
author = {Arcozzi, Nicola and Chalmoukis, Nikolaos},
title = {Riesz capacities of a set due to {Dobi\'nski}},
journal = {Comptes Rendus. Math\'ematique},
pages = {679--685},
publisher = {Acad\'emie des sciences, Paris},
volume = {360},
number = {G6},
year = {2022},
doi = {10.5802/crmath.332},
zbl = {07547266},
language = {en},
url = {http://www.numdam.org/articles/10.5802/crmath.332/}
}
TY - JOUR AU - Arcozzi, Nicola AU - Chalmoukis, Nikolaos TI - Riesz capacities of a set due to Dobiński JO - Comptes Rendus. Mathématique PY - 2022 SP - 679 EP - 685 VL - 360 IS - G6 PB - Académie des sciences, Paris UR - http://www.numdam.org/articles/10.5802/crmath.332/ DO - 10.5802/crmath.332 LA - en ID - CRMATH_2022__360_G6_679_0 ER -
%0 Journal Article %A Arcozzi, Nicola %A Chalmoukis, Nikolaos %T Riesz capacities of a set due to Dobiński %J Comptes Rendus. Mathématique %D 2022 %P 679-685 %V 360 %N G6 %I Académie des sciences, Paris %U http://www.numdam.org/articles/10.5802/crmath.332/ %R 10.5802/crmath.332 %G en %F CRMATH_2022__360_G6_679_0
Arcozzi, Nicola; Chalmoukis, Nikolaos. Riesz capacities of a set due to Dobiński. Comptes Rendus. Mathématique, Tome 360 (2022) no. G6, pp. 679-685. doi : 10.5802/crmath.332. http://www.numdam.org/articles/10.5802/crmath.332/
[1] Function spaces and potential theory, Grundlehren der Mathematischen Wissenschaften, 314, Springer, 1996 | DOI
[2] A trigonometric infinite product, Am. Math. Mon., Volume 54 (1947) no. 4, pp. 206-211 | DOI | MR | Zbl
[3] Two-weight dyadic Hardy’s inequalities (2021) | arXiv
[4] Potential theory on trees, graphs and Ahlfors-regular metric spaces, Potential Anal., Volume 41 (2013) no. 2, pp. 317-366 | DOI | MR | Zbl
[5] Random walks on a tree and capacity in the interval, Ann. Inst. Henri Poincaré, Probab. Stat., Volume 28 (1992) no. 4, pp. 557-592 | Numdam | MR | Zbl
[6] A mass transference principle and the Duffin–Schaeffer conjecture for Hausdorff measures, Ann. Math., Volume 164 (2006) no. 3, pp. 971-992 | DOI | MR | Zbl
[7] On a characterization of bilinear forms on the Dirichlet space, Proc. Am. Math. Soc., Volume 140 (2012) no. 7, pp. 2429-2440 | DOI | MR | Zbl
[8] Totally null sets and capacity in Dirichlet type spaces, J. Lond. Math. Soc. (2022) (in Early View) | DOI
[9] Some remarks on the Dirichlet problem on infinite trees, Concrete Operators, Volume 6 (2019) no. 1, pp. 20-32 | DOI | MR | Zbl
[10] Hausdorff measures, dyadic approximations, and the Dobiński set, Ill. J. Math., Volume 65 (2021) no. 2, pp. 515-531 | DOI | Zbl
[11] Product einer unendlichen Factorenreihe, Archiv der Mathematik und Physik, Volume 59 (1876), pp. 98-100
[12] Producte einiger Factorenreihen, Archiv der Mathematik und Physik, Volume 61 (1877), pp. 434-438 | Zbl
[13] Phase transition of logarithmic capacity for the uniform -sets, Potential Anal., Volume 56 (2021) no. 4, pp. 597-622 | DOI | MR | Zbl
[14] On numbers badly approximable by dyadic rationals, Isr. J. Math., Volume 171 (2009) no. 1, pp. 93-110 | DOI | MR | Zbl
[15] Potential theory on infinite networks, Lecture Notes in Mathematics, 1590, Springer, 1994 | DOI
Cité par Sources :
