Without assuming the Northcott property we provide an upper bound on the number of “big solutions” of a special system of Diophantine inequalities over proper adelic curves. This system is interesting since it is a stronger version of Roth’s inequality for adelic curves.
Sans supposer la propriété de Northcott, nous donnons une majoration pour le nombre de « grandes solutions » d’un système spécial d’inégalités diophantiennes sur les courbes adéliques propres. Ce système est intéressant car il s’agit d’une version renforcée de l’inégalité de Roth pour les courbes adéliques.
Révisé le :
Accepté le :
Publié le :
Keywords: Diophantine Inequalities, Roth’s Theorem, Adelic Curves
Paolo Dolce 1
CC-BY-ND 4.0
@article{JTNB_2025__37_1_143_0,
author = {Paolo Dolce},
title = {A note on some {Diophantine} inequalities over adelic curves},
journal = {Journal de th\'eorie des nombres de Bordeaux},
pages = {143--152},
year = {2025},
publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
volume = {37},
number = {1},
doi = {10.5802/jtnb.1316},
language = {en},
url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1316/}
}
TY - JOUR AU - Paolo Dolce TI - A note on some Diophantine inequalities over adelic curves JO - Journal de théorie des nombres de Bordeaux PY - 2025 SP - 143 EP - 152 VL - 37 IS - 1 PB - Société Arithmétique de Bordeaux UR - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1316/ DO - 10.5802/jtnb.1316 LA - en ID - JTNB_2025__37_1_143_0 ER -
%0 Journal Article %A Paolo Dolce %T A note on some Diophantine inequalities over adelic curves %J Journal de théorie des nombres de Bordeaux %D 2025 %P 143-152 %V 37 %N 1 %I Société Arithmétique de Bordeaux %U https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1316/ %R 10.5802/jtnb.1316 %G en %F JTNB_2025__37_1_143_0
Paolo Dolce. A note on some Diophantine inequalities over adelic curves. Journal de théorie des nombres de Bordeaux, Tome 37 (2025) no. 1, pp. 143-152. doi: 10.5802/jtnb.1316
[1] Some quantitative results related to Roth’s theorem, J. Aust. Math. Soc., Ser. A, Volume 45 (1988) no. 2, pp. 233-248 | DOI | Zbl | MR
[2] Corrigenda: “Some quantitative results related to Roth’s theorem”, J. Aust. Math. Soc., Ser. A, Volume 48 (1990) no. 1, pp. 154-155 | DOI | Zbl | MR
[3] Bounds for the solutions of unit equations, Acta Arith., Volume 74 (1996) no. 1, pp. 67-80 | DOI | MR | Zbl
[4] Arakelov geometry over Adelic curves, Lecture Notes in Mathematics, 2258, Springer, 2020 | DOI | Zbl | MR
[5] Arithmetic intersection theory over adelic curves (2021) | arXiv | Zbl
[6] Autour du théorème de Roth, Monatsh. Math., Volume 124 (1997) no. 2, pp. 147-175 | Zbl | DOI | MR
[7] Rational approximations to algebraic numbers, Mathematika, Volume 2 (1955), pp. 160-167 | Zbl | DOI | MR
[8] On the generalisation of Roth’s theorem (2023) (in press in Kyoto J. Math., preprint available at https://arxiv.org/abs/2111.12409)
[9] Dyson’s lemma for polynomials in several variables (and the theorem of Roth), Invent. Math., Volume 78 (1984) no. 3, pp. 445-490 | Zbl | DOI | MR
[10] On equations in -units and the Thue–Mahler equation, Invent. Math., Volume 75 (1984) no. 3, pp. 561-584 | Zbl | DOI | MR
[11] A note on Roth’s theorem, J. Number Theory, Volume 36 (1990) no. 1, pp. 127-132 | Zbl | DOI | MR
[12] The number of exceptional approximations in Roth’s theorem, J. Aust. Math. Soc., Ser. A, Volume 59 (1995) no. 3, pp. 375-383 | DOI | Zbl | MR
[13] Roth’s Theorem over arithmetic function fields, Algebra Number Theory, Volume 15 (2021) no. 8, pp. 1943-2017 | DOI | MR | Zbl
[14] Lecture notes on Diophantine analysis, Appunti. Scuola Normale Superiore di Pisa (Nuova Serie), 8, Edizioni della Normale, 2009, xvi+237 pages (with an appendix by Francesco Amoroso) | MR | Zbl
Cité par Sources :