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Note on a Theorem of Myrberg

Published online by Cambridge University Press:  24 October 2008

J. Gillis
J. Gillis
Affiliation:
Trinity College
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Extract

1. Prof. Fekete, in a letter to the author, wrote: “…Satz von Myrberg über den Zusammenhang zwischen dem transfiniten Durchmesser und dem logarithmischen Mass zugehörig zum Exponenten 1 + ε, ε > 0 (vgl. Satz 10 in seiner Acta Arbeit).…Ich würde einen elementaren Beweis für den letzteren Satz sehr begrüssen.” The object of this note is to supply such a proof but, since the concepts involved may not all be matters of common knowledge, I shall begin by outlining the necessary definitions, the relevant known results, the nature of the theorem and its relation to what is known.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 33 , Issue 4 , October 1937 , pp. 419 - 424
Copyright
Copyright © Cambridge Philosophical Society 1937

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References

† Cf. Fekete, M., “Über den Transfiniten Durchmesser ebener Punktmengen”, Math. Zeitschrift, 32 (1930), 108–114.CrossRefGoogle Scholar

‡ Cf. Hausdorff, , “Dimension und äusseres Mass”, Math. Annalen, 79 (1919), 157–179.CrossRefGoogle Scholar

* Erdös, P. and Gillis, J., “Note on the transfinite diameter”, Journ. Lond. Math. Soc. (in the press).Google Scholar

† In a paper to appear in the Journal of the London Mathematical Society.

‡ Myrberg, P. J., “Die Greensche Funktionen für eine gegebene Riemannsche Fläche”, Acta Mathematica, 61 (1933), 39–79CrossRefGoogle Scholar; see in particular theorem 10, p. 63.

§ Nevanlinna, R., “Über die Kapazität der Cantorschen Punktmengen”, Monatshefte für Math, und Phys. 43 (1936), 435–447.CrossRefGoogle Scholar

* We use ø(E) to denote the set of values taken by ø(x) as x ranges through E; similarly for ø(I), Σø(Ir), etc.

† Besicovitch, A. S., “On linear sets of points of fractional dimension”, Math. Annalen, 101 (1929), 161–193CrossRefGoogle Scholar. By the right-hand upper density we mean, of course, and similarly for the left-hand upper density.

* This is obtained by estimating the area in question by integration with respect to w. The argument is similar to the “intuitive proof” of Young's inequality; cf. Hardy, , Littlewood, and Pólya, , Inequalities (Cambridge, 1934), p. 111.Google Scholar

† The necessity of proving this last result should not occasion surprise since, on the basis of the hypothesis of the continuum, it is actually false for general measurable sets. See Besicovitch, A. S., “Most concentrated and most rarified sets”, Acta Mathematica, 62 (1934), 289–300CrossRefGoogle Scholar; cf. also Szpilgin, E., “Sur un ensemble non-mesurable de M. Sierpinski”, C.R. Soc. Sc. Varsovie, 24 (1931), 78–85Google Scholar, and Sierpinski, , Hypothèse du continu (Warsaw, 1934), p. 92.Google Scholar