On Absolute and Quantitative Subspace Theorems
-
Hieu T. Ngo
and
Si Duc Quang
Abstract
The Absolute Subspace Theorem, a vast generalization and a quantitative improvement of Schmidt’s Subspace Theorem, was first established by Evertse and Schlickewei and then strengthened remarkably by Evertse and Ferretti. We study quantitative generalizations and extensions of subspace theorems in various contexts. We establish a generalization of Evertse and Ferretti’s Absolute Subspace Theorem for hyperplanes in general position. We obtain improved (non-absolute) Quantitative Subspace Theorems for hyperplanes in general position and in subgeneral position. We show a Semi-quantitative Subspace Theorem for hyperplanes in non-subdegenerate position.
Funding statement: The research of Hieu T. Ngo is supported by Vietnam Academy of Science and Technology (VAST) under the grant number THTETN.08/22-24, and is funded by Vingroup Joint Stock Company and supported by Vingroup Innovation Foundation (VinIF) under the project code VINIF.2021.DA00030. This work was done during a stay of Si Duc Quang at Vietnam Institute for Advanced Study in Mathematics (VIASM). Si Duc Quang would like to thank the staff there, as well as the partial support of VIASM.
Acknowledgements
We would like to thank the anonymous referee for a careful reading of the paper and for many helpful suggestions that helped improve the presentation of the paper. The application of Theorem 21 to unit equations (Theorem 22) was kindly suggested to the authors by the referee.
References
[1] E. Bombieri and W. Gubler, Heights in Diophantine Geometry, New Math. Monogr. 4, Cambridge University, Cambridge, 2006. Search in Google Scholar
[2] Y. Bugeaud, Quantitative versions of the subspace theorem and applications, J. Théor. Nombres Bordeaux 23 (2011), no. 1, 35–57. 10.5802/jtnb.749Search in Google Scholar
[3] Z. Chen and M. Ru, Integer solutions to decomposable form inequalities, J. Number Theory 115 (2005), no. 1, 58–70. 10.1016/j.jnt.2004.10.004Search in Google Scholar
[4] J.-H. Evertse, On sums of S-units and linear recurrences, Compos. Math. 53 (1984), no. 2, 225–244. Search in Google Scholar
[5] J.-H. Evertse, An improvement of the quantitative subspace theorem, Compos. Math. 101 (1996), no. 3, 225–311. Search in Google Scholar
[6] J.-H. Evertse, On the quantitative subspace theorem, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 377 (2010), no. 10, 217–240, 245. Search in Google Scholar
[7] J.-H. Evertse and R. G. Ferretti, Diophantine inequalities on projective varieties, Int. Math. Res. Not. IMRN 2002 (2002), no. 25, 1295–1330. 10.1155/S107379280210804XSearch in Google Scholar
[8] J.-H. Evertse and R. G. Ferretti, A generalization of the subspace theorem with polynomials of higher degree, Diophantine Approximation, Dev. Math. 16, Springer, Vienna (2008), 175–198. 10.1007/978-3-211-74280-8_9Search in Google Scholar
[9] J.-H. Evertse and R. G. Ferretti, A further improvement of the Quantitative Subspace Theorem, Ann. of Math. (2) 177 (2013), no. 2, 513–590. 10.4007/annals.2013.177.2.4Search in Google Scholar
[10] J.-H. Evertse and K. Györy, Unit Equations in Diophantine Number Theory, Cambridge Stud. Adv. Math. 146, Cambridge University, Cambridge, 2015. 10.1017/CBO9781316160749Search in Google Scholar
[11] J.-H. Evertse, K. Györy, C. L. Stewart and R. Tijdeman, S-unit equations and their applications, New Advances In Transcendence Theory, Cambridge University, Cambridge (1988), 110–174. 10.1017/CBO9780511897184.010Search in Google Scholar
[12] J.-H. Evertse and H. P. Schlickewei, The Absolute Subspace Theorem and linear equations with unknowns from a multiplicative group, Number Theory in Progress, Vol. 1 (Zakopane-Kościelisko 1997), De Gruyter, Berlin (1999), 121–142. 10.1515/9783110285581.121Search in Google Scholar
[13] J.-H. Evertse and H. P. Schlickewei, A quantitative version of the Absolute Subspace Theorem, J. Reine Angew. Math. 548 (2002), 21–127. 10.1515/crll.2002.060Search in Google Scholar
[14] M. Hindry and J. H. Silverman, Diophantine Geometry. An Introduction, Grad. Texts in Math. 201, Springer, New York, 2000. 10.1007/978-1-4612-1210-2Search in Google Scholar
[15] N. Hirata-Kohno, An application of quantitative subspace theorem, Analytic Number Theory, Kyoto University, Kyoto (1994), 81–87. Search in Google Scholar
[16] Y. Liu, On the problem of integer solutions to decomposable form inequalities, Int. J. Number Theory 4 (2008), no. 5, 859–872. 10.1142/S1793042108001766Search in Google Scholar
[17] J. Noguchi and J. Winkelmann, Nevanlinna Theory in Several Complex Variables and Diophantine Approximation, Grundlehren Math. Wiss. 350, Springer, Tokyo, 2014. 10.1007/978-4-431-54571-2Search in Google Scholar
[18] D. H. Pham, Schmidt’s subspace theorem for non-subdegenerate families of hyperplanes, Int. J. Number Theory 18 (2022), no. 3, 557–574. 10.1142/S1793042122500312Search in Google Scholar
[19] S. D. Quang, Degeneracy second main theorem for meromorphic mappings and moving hypersurfaces with truncated counting functions and applications, Internat. J. Math. 31 (2020), no. 6, Article ID 2050045. 10.1142/S0129167X20500457Search in Google Scholar
[20] K. F. Roth, Rational approximations to algebraic numbers, Mathematika 2 (1955), 1–20. 10.1112/S0025579300000644Search in Google Scholar
[21]
M. Ru and P.-M. Wong,
Integral points of
[22] W. M. Schmidt, Norm form equations, Ann. of Math. (2) 96 (1972), 526–551. 10.2307/1970824Search in Google Scholar
[23] W. M. Schmidt, Diophantine Approximation, Lecture Notes in Math. 785, Springer, Berlin, 1980. Search in Google Scholar
[24] W. M. Schmidt, The subspace theorem in Diophantine approximations, Compos. Math. 69 (1989), no. 2, 121–173. Search in Google Scholar
[25] W. M. Schmidt, Diophantine Approximations and Diophantine Equations, Lecture Notes in Math. 1467, Springer, Berlin, 1991. 10.1007/BFb0098246Search in Google Scholar
[26] P. Vojta, On the Nochka–Chen–Ru–Wong proof of Cartan’s conjecture, J. Number Theory 125 (2007), no. 1, 229–234. 10.1016/j.jnt.2006.10.014Search in Google Scholar
[27] P. Vojta, Diophantine approximation and Nevanlinna theory, Arithmetic Geometry, Lecture Notes in Math. 2009, Springer, Berlin (2011), 111–224. 10.1007/978-3-642-15945-9_3Search in Google Scholar
© 2024 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Frontmatter
- Square-integrable representations and the coadjoint action of solvable Lie groups
- Deviation probabilities for extremal eigenvalues of large Chiral non-Hermitian random matrices
- Nef vector bundles on a hyperquadric with first Chern class two
- Scaling spectrum of a class of self-similar measures with product form on ℝ
- Idempotent decomposition of regularity and characterization for the accumulation of associated spectrum
- Geodesic orbit Randers metrics in homogeneous bundles over generalized Stiefel manifolds
- Proof of some conjectures of Guo and of Tang
- On Absolute and Quantitative Subspace Theorems
- Birational equivalence of the Zassenhaus varieties for basic classical Lie superalgebras and their purely-even reductive Lie subalgebras in odd characteristic
- Geometric approach to the Moore–Penrose inverse and the polar decomposition of perturbations by operator ideals
- The Weil bound for generalized Kloosterman sums of half-integral weight
- Relative Rota–Baxter groups and skew left braces
- Asai gamma factors over finite fields
- Representations of non-finitely graded Lie algebras related to Virasoro algebra
- Multiple solutions for fractional elliptic systems
- Arithmetic Bohr radius for the Minkowski space
- On Strichartz estimates for many-body Schrödinger equation in the periodic setting
Articles in the same Issue
- Frontmatter
- Square-integrable representations and the coadjoint action of solvable Lie groups
- Deviation probabilities for extremal eigenvalues of large Chiral non-Hermitian random matrices
- Nef vector bundles on a hyperquadric with first Chern class two
- Scaling spectrum of a class of self-similar measures with product form on ℝ
- Idempotent decomposition of regularity and characterization for the accumulation of associated spectrum
- Geodesic orbit Randers metrics in homogeneous bundles over generalized Stiefel manifolds
- Proof of some conjectures of Guo and of Tang
- On Absolute and Quantitative Subspace Theorems
- Birational equivalence of the Zassenhaus varieties for basic classical Lie superalgebras and their purely-even reductive Lie subalgebras in odd characteristic
- Geometric approach to the Moore–Penrose inverse and the polar decomposition of perturbations by operator ideals
- The Weil bound for generalized Kloosterman sums of half-integral weight
- Relative Rota–Baxter groups and skew left braces
- Asai gamma factors over finite fields
- Representations of non-finitely graded Lie algebras related to Virasoro algebra
- Multiple solutions for fractional elliptic systems
- Arithmetic Bohr radius for the Minkowski space
- On Strichartz estimates for many-body Schrödinger equation in the periodic setting
