On integer points in polygons
Annales de l'Institut Fourier, Tome 43 (1993) no. 2, pp. 313-323.

Le phénomène de termes d’erreur anormalement petits dans le problème des points entiers dans un polygone est étudié en dimension 2. Pour des polygones irrationnels, les erreurs sont exprimées en termes de propriétés diophantiennes des pentes des côtés. Il en résulte pour le nombre de points entiers dans le dilaté de rapport t,t→∞, de certaines classes de polygones irrationnels que le terme d’erreur est borné ≪ℓnq avec q>0 ou comme ≪tε avec ε>0 arbitraire.

The phenomenon of anomaly small error terms in the lattice point problem is considered in detail in two dimensions. For irrational polygons the errors are expressed in terms of diophantine properties of the side slopes. As a result, for the t-dilatation, t→∞, of certain classes of irrational polygons the error terms are bounded as ≪ℓnqt with some q>0, or as ≪tε with arbitrarily small ε>0.

@article{AIF_1993__43_2_313_0,
     author = {Skriganov, Maxim},
     title = {On integer points in polygons},
     journal = {Annales de l'Institut Fourier},
     pages = {313--323},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {43},
     number = {2},
     year = {1993},
     doi = {10.5802/aif.1333},
     mrnumber = {94d:11077},
     zbl = {0779.11041},
     language = {en},
     url = {https://www.numdam.org/articles/10.5802/aif.1333/}
}
TY  - JOUR
AU  - Skriganov, Maxim
TI  - On integer points in polygons
JO  - Annales de l'Institut Fourier
PY  - 1993
SP  - 313
EP  - 323
VL  - 43
IS  - 2
PB  - Institut Fourier
PP  - Grenoble
UR  - https://www.numdam.org/articles/10.5802/aif.1333/
DO  - 10.5802/aif.1333
LA  - en
ID  - AIF_1993__43_2_313_0
ER  - 
%0 Journal Article
%A Skriganov, Maxim
%T On integer points in polygons
%J Annales de l'Institut Fourier
%D 1993
%P 313-323
%V 43
%N 2
%I Institut Fourier
%C Grenoble
%U https://www.numdam.org/articles/10.5802/aif.1333/
%R 10.5802/aif.1333
%G en
%F AIF_1993__43_2_313_0
Skriganov, Maxim. On integer points in polygons. Annales de l'Institut Fourier, Tome 43 (1993) no. 2, pp. 313-323. doi : 10.5802/aif.1333. https://www.numdam.org/articles/10.5802/aif.1333/

[CV] Y. Colin De Verdière, Nombre de points entiers dans une famille homothétique de domaines de ℝn, Ann. Sci. École Norm. Sup., 4e série, 10 (1977), 559-576. | Numdam | Zbl

[HL] G. H. Hardy, J.E. Littlewood, Some problems of Diophantine approximation : the lattice points of a right-angled triangle, part I, Proc. London Math. Soc. (2), 20 (1922), 15-36; part II, Abh. Math. Sem. Hamburg, 1 (1922), 212-249. | JFM

[Kh] A. Khinchin, Einige Sätze über Kettenbrüche, mit Anwendungen auf die Theorie der Diophantischen Approximationen, Math. Ann., 92 (1924), 115-125. | JFM

[KN] I. Kuipers, H. Niederreiter, Uniform distribution of sequences, Wiley, New-York-London, 1974. | Zbl

[L] S. Lang, Introduction to diophantine approximations, Addison-Wesley, Mass., 1966. | MR | Zbl

[P] O. Perron, Die Lehre von den Kettenbrüchen, 3 Aufl., Teubner, Stuttgart, 1954. | Zbl

[R1] B. Randol, A lattice point problem I, Trans. A.M.S., 121 (1966), 257-268 ; II, Trans. A.M.S., 125 (1966), 101-113. | Zbl

[R2] B. Randol, On the Fourier transform of the indicator function of a planar set, Trans. A.M.S., 139 (1969), 271-278. | MR | Zbl

[Sch] W.M. Schmidt, Diophantine approximation, Lecture Notes in Math., 785, Springer-Verlag, Berlin, New York, 1980. | MR | Zbl

[S1] M.M. Skriganov, On lattices in algebraic number fields, Dokl. Akad. Nauk SSSR, 306 (1989), 553-555, Soviet Math. Dokl., 39 (1989), 538-540. | MR | Zbl

[S2] M.M. Skriganov, Lattices in algebraic number fields and uniform distributions modulo 1, LOMI Preprint 12-88, Leningrad, (1988), Algebra and analysis, 1, N2 (1989), 207-228, Leningrad Math. J., 1, N2 (1990), 535-558. | Zbl

[S3] M.M. Skriganov, Construction of uniform distributions in terms of geometry of numbers, Prépublication de l'Institut Fourier, n° 200, Grenoble, 1992.

[S4] M.M. Skriganov, Anomaly small errors in the lattice point problem, (in preparation).

  • Cherubini, Giacomo; Yatsyna, Pavlo Cyclic Polytope of the Simplest Cubic Fields, Discrete Computational Geometry, Volume 69 (2023) no. 2, p. 338 | DOI:10.1007/s00454-021-00369-2
  • Borda, Bence Lattice Points in Algebraic Cross-polytopes and Simplices, Discrete Computational Geometry, Volume 60 (2018) no. 1, p. 145 | DOI:10.1007/s00454-017-9946-z
  • Narkiewicz, Władysław The Twenties, Rational Number Theory in the 20th Century (2012), p. 131 | DOI:10.1007/978-0-85729-532-3_3
  • Nechayeva, Marina; Randol, Burton Asymptotics of Weighted Lattice Point Counts Inside Dilating Polygons, Additive Number Theory (2010), p. 287 | DOI:10.1007/978-0-387-68361-4_20
  • Travaglini, Giancarlo Average Decay of the Fourier Transform, Fourier Analysis and Convexity (2004), p. 245 | DOI:10.1007/978-0-8176-8172-2_11
  • Kleinbock, Dmitry; Shah, Nimish; Starkov, Alexander Chapter 11 Dynamics of subgroup actions on homogeneous spaces of lie groups and applications to number theory, Volume 1 (2002), p. 813 | DOI:10.1016/s1874-575x(02)80013-3
  • Moshchevitina, L. A. On integer points in polygons, Mathematical Notes, Volume 65 (1999) no. 4, p. 525 | DOI:10.1007/bf02675371
  • Мощевитина, Л А; Moshchevitina, L A О целых точках в многоугольниках, Математические заметки, Volume 65 (1999) no. 4, p. 629 | DOI:10.4213/mzm1092
  • Brandolini, Luca; Colzani, Leonardo; Travaglini, Giancarlo Average decay of Fourier transforms and integer points in polyhedra, Arkiv för Matematik, Volume 35 (1997) no. 2, p. 253 | DOI:10.1007/bf02559969

Cité par 9 documents. Sources : Crossref