Riesz capacities of a set due to Dobiński

Nicola Arcozzi; Nikolaos Chalmoukis

doi:10.5802/crmath.332

Analyse harmonique, Combinatoire

Riesz capacities of a set due to Dobiński

Nicola Arcozzi ¹ ; Nikolaos Chalmoukis ¹

¹ Dipartimento di Matematica, Università di Bologna, 40126, Bologna, Italy

Comptes Rendus. Mathématique, Volume 360 (2022), pp. 679-685.

Résumé

We study the Riesz $(a, p)$ -capacity of the so called Dobiński set. We characterize the values of the parameters $a$ and $p$ for which the $(a, p)$ -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 $(a, p)$ -capacities which seem to be better adapted to the problem.

Reçu le : 2021-11-11
Révisé le : 2022-01-14
Accepté le : 2022-01-18
Publié le : 2022-06-22

Zbl

DOI : 10.5802/crmath.332

Classification : 31C20, 30C85, 31A15, 11J83
Mots clés : Riesz capacity, Logarithmic capacity, Dobiński set, Dyadic capacity, Non-linear capacity, Diophantine approxmation

Affiliations des auteurs :

Nicola Arcozzi ¹ ; Nikolaos Chalmoukis ¹

¹ Dipartimento di Matematica, Università di Bologna, 40126, Bologna, Italy

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {Riesz capacities of a set due to {Dobi\'nski}},
     journal = {Comptes Rendus. Math\'ematique},
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     doi = {10.5802/crmath.332},
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     language = {en},
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Nicola Arcozzi; Nikolaos Chalmoukis. Riesz capacities of a set due to Dobiński. Comptes Rendus. Mathématique, Volume 360 (2022), pp. 679-685. doi : 10.5802/crmath.332. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.332/

Bibliographie
Cité par

[1] David R. Adams; Lars I. Hedberg Function spaces and potential theory, Grundlehren der Mathematischen Wissenschaften, 314, Springer, 1996 | DOI

[2] Ralph P. Agnew; Robert J. Walker A trigonometric infinite product, Am. Math. Mon., Volume 54 (1947) no. 4, pp. 206-211 | DOI | MR | Zbl

[3] Nicola Arcozzi; Nikolaos Chalmoukis; Matteo Levi; Pavel Mozolyako Two-weight dyadic Hardy’s inequalities (2021) | arXiv

[4] Nicola Arcozzi; Richard Rochberg; Eric T. Sawyer; Brett D. Wick Potential theory on trees, graphs and Ahlfors-regular metric spaces, Potential Anal., Volume 41 (2013) no. 2, pp. 317-366 | DOI | MR | Zbl

[5] Itai Benjamini; Yuval Peres 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] Victor Beresnevich; Sanju Velani 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] Carme Cascante; Joaquin M. Ortega 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] Nikolaos Chalmoukis; Michael Hartz Totally null sets and capacity in Dirichlet type spaces, J. Lond. Math. Soc. (2022) (in Early View) | DOI

[9] Nikolaos Chalmoukis; Matteo Levi Some remarks on the Dirichlet problem on infinite trees, Concrete Operators, Volume 6 (2019) no. 1, pp. 20-32 | DOI | MR | Zbl

[10] Alberto Dayan; José L. Fernández; María J. González Hausdorff measures, dyadic approximations, and the Dobiński set, Ill. J. Math., Volume 65 (2021) no. 2, pp. 515-531 | DOI | Zbl

[11] G. Dobinśki Product einer unendlichen Factorenreihe, Archiv der Mathematik und Physik, Volume 59 (1876), pp. 98-100

[12] G. Dobinśki Producte einiger Factorenreihen, Archiv der Mathematik und Physik, Volume 61 (1877), pp. 434-438 | Zbl

[13] Victor Kleptsyn; Fernando Quintino Phase transition of logarithmic capacity for the uniform $G_{δ}$ -sets, Potential Anal., Volume 56 (2021) no. 4, pp. 597-622 | DOI | MR | Zbl

[14] Johan Nilsson On numbers badly approximable by dyadic rationals, Isr. J. Math., Volume 171 (2009) no. 1, pp. 93-110 | DOI | MR | Zbl

[15] Paolo Soardi Potential theory on infinite networks, Lecture Notes in Mathematics, 1590, Springer, 1994 | DOI

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