Hostname: page-component-848d4c4894-sjtt6 Total loading time: 0 Render date: 2024-06-16T01:45:25.518Z Has data issue: false hasContentIssue false
  • English
  • Français

An explicit dense universal Hilbert set

Published online by Cambridge University Press:  20 June 2018

MICHAEL FILASETA  and
ROBERT WILCOX
MICHAEL FILASETA
Affiliation:
Department of Mathematics, LeConte College, 1523 Greene Street, University of South Carolina, Columbia, SC 29208U.S.A. e-mails: filaseta@math.sc.edu, rwilcox5@gmail.com
ROBERT WILCOX
Affiliation:
Department of Mathematics, LeConte College, 1523 Greene Street, University of South Carolina, Columbia, SC 29208U.S.A. e-mails: filaseta@math.sc.edu, rwilcox5@gmail.com
Get access Rights & Permissions [Opens in a new window]

Abstract

We provide the first explicit example of a universal Hilbert set ${\Ncal S}$ having asymptotic density 1 in the set of integers. More precisely, the number of integers not in ${\Ncal S}$ with absolute value ≤ X is bounded by X/(log X)δ, where δ = 1 − (1 + loglog 2)/(log 2) = 0.086071. . ..

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 167 , Issue 3 , November 2019 , pp. 531 - 547
Copyright
Copyright © Cambridge Philosophical Society 2018 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1] Bilu, Y. A note on universal Hilbert sets. J. Reine Angew. Math. 479 (1996), 195–203.Google Scholar
[2] Bilu, Y. The many faces of the subspace theorem [after Adamczewski, Bugeaud, Corvaja, Zannier. . .], Séminaire Bourbaki. Vol. 2006/2007. Astérisque, No. 317 (2008), Exp. No. 967, vii, 1–38.Google Scholar
[3] Cauchy, A. L. Résolution des équations numériques et sur la théorie de l'élimination. Exercices de Mathématiques, Œuvres Complétes D'Augustin Cauchy, Série 2, 9 (1891), 87–161 (Gauthier-Villars et Fils).Google Scholar
[4] Corvaja, P. and Zannier, U. Diophantine equations with power sums and universal Hilbert sets, Indag. Math. (N.S.) 9 (1998), 317–332.Google Scholar
[5] Dèbes, P. and Zannier, U. Universal Hilbert subsets, Math. Proc. Camb. Phil. Soc. 124 (1998), 127–134.Google Scholar
[6] Ford, K. The distribution of integers with a divisor in a given interval, Ann. of Math. (2) 168 (2008), 367–433.Google Scholar
[7] Fried, M. D. and Jarden, M. Field arithmetic. Ergeb. Math. Grenzgeb. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], Vol. 11, third Edition (Springer-Verlag, Berlin, 2008).Google Scholar
[8] Gilmore, P. C. and Robinson, A. Metamathematical considerations on the relative irreducibility of polynomials. Canad. J. Math. 7 (1955), 483–489.Google Scholar
[9] Hadlock, C. R. Field Theory and Its Classical Problems. Carus Math. Monogr. Math. Assoc. of America (Washington, DC, 1978).Google Scholar
[10] Hardy, G. H. and Wright, E. M. An Introduction to the Theory of Numbers. Fifth Edition (The Clarendon Press, Oxford University Press, New York, 1979).Google Scholar
[11] Hilbert, D. Ueber die Irreducibilität ganzer rationaler Functionen mit ganzzahligen Coefficienten. J. Reine Angew. Math. 110 (1892), 104–129.Google Scholar
[12] Lang, S. Fundamentals of Diophantine Geometry (Springer-Verlag, New York, 1983).Google Scholar
[13] Schinzel, A. Selected Topics on Polynomials (University of Michigan Press, Ann Arbor, Mich., 1982).Google Scholar
[14] Schinzel, A. Polynomials with Special Regard to Reducibility, with an appendix by Umberto Zannier. Encyclopedia of Mathematics and its Applications, Vol. 77 (Cambridge University Press, Cambridge, 2000).Google Scholar
[15] Schinzel, A. and Zannier, U. The least admissible value of the parameter in Hilbert's irreducibility theorem. Acta Arith. 69 (1995), 293–302.Google Scholar
[16] Serre, J.-P.. Topics in Galois Theory. Research Notes in Mathematics, Vol. 1, second Edition. With notes by Henri Darmon (A K Peters, Ltd., Wellesley, MA, 2008).Google Scholar
[17] Siegel, C. L. Über einige Anwendungen Diophantischer Approximationen. Abh. Preuss Akad. Wiss. Phys.-Math. Kl. 1 (1929), 1–70.Google Scholar
[18] Sprindžuk, V. G. Diophantine equations involving unknown prime numbers. Trudy Math. Inst. Steklov 158 (1981), 180–196; English transl. in Proc. Steklov Inst. Math. (1983), Issue 4, 197–214.Google Scholar
[19] Zannier, U. Note on dense universal Hilbert sets. C. R. Acad. Sci. Paris Sér. I Math. 322 (1996), 703–706.Google Scholar