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The real cubic case of Mahler's conjecture

  • B. Volkmann (a1)

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For any (real or complex) transcendental number ξ and any integer n > 0 let ϑn(ξ) be the least upper bound of the set of all positive numbers σ for which there exist infinitely many polynomials p1(x), p2(x), … of degree n, with integer coefficients, satisfying

where ‖pi‖ denotes the “height” of pi(x), i.e. the maximum modulus of the coefficients. Plainly ϑn(ξ) serves as a measure of how well (or how badly) the number zero can be approximated by values of nth degree integral polynomials at the point ξ. It can be shown by means of the “Schubfachprinzip” that, at worst,

if the transcendental number ξ is real, and

if it is complex, i.e.ϑn(ξ) is never smaller than these bounds. Furthermore, a conjecture of K. Mahler may be interpreted as stating that for almost all real and for almost all complex numbers the equations (2) and (3), respectively, are actually true; in other words, almost all transcendental numbers have the worst possible approximation property for any degree n.

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1.Davenport, H., “A note on binary cubic forms”, Mathematika, 8 (1961), 5862.
2.Kasch, F., “Über eine metrische Eigenschaft dor S-Zahlen”, Math. Zeit., 70 (1958), 263270.
3.Kasch, F., “Ein metrischer Beitrag über Mahlersche S-Zahlen. II”, Journ. reine angew. Math., 203 (1960), 157159.
4.Kasch, F., and Volkmann, B., “Zur Mahlerschen Vermutung über S-Zahlen”, Math. Ann. 136 (1958), 442453.
5.Kubilyus, J. F., “On the application of Vinogradov's method to the solution of a problem in motric number theory”, Dolk. Akad. Nauk USSR, N.S., 67 (1949), 783786 (in Russian).
6.Schneider, Th., Einführung in die transzendenten Zahlen (Springer, Berlin-Göttingen-Heidelberg, 1957).
7.Volkmann, B., “Zum kubischen Fall der Mahlerschen Vermutung”, Math. Ann., 139 (1959), 8790.
8.Volkmann, B., “Ein metrischer Beitrag über Mahlersche S-Zahlen. I”. Journ. reine angew. Math., 203 (1960), 154156.
9.Volkmann, B., “Zur Mahlerschen Vermutung im Komplexen”, Math. Ann., 140 (1960), 351359.
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The real cubic case of Mahler's conjecture

  • B. Volkmann (a1)

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