On integer points in polygons
Annales de l'Institut Fourier, Volume 43 (1993) no. 2, p. 313-323
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 n q t with some q>0, or as t ε with arbitrarily small ε>0.
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 n q avec q>0 ou comme t ε avec ε>0 arbitraire.
@article{AIF_1993__43_2_313_0,
     author = {Skriganov, Maxim},
     title = {On integer points in polygons},
     journal = {Annales de l'Institut Fourier},
     publisher = {Imprimerie Louis-Jean},
     address = {Gap},
     volume = {43},
     number = {2},
     year = {1993},
     pages = {313-323},
     doi = {10.5802/aif.1333},
     zbl = {0779.11041},
     mrnumber = {94d:11077},
     language = {en},
     url = {http://www.numdam.org/item/AIF_1993__43_2_313_0}
}
Skriganov, Maxim. On integer points in polygons. Annales de l'Institut Fourier, Volume 43 (1993) no. 2, pp. 313-323. doi : 10.5802/aif.1333. http://www.numdam.org/item/AIF_1993__43_2_313_0/

[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 0409.58011

[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 48.0197.07

[Kh] A. Khinchin, Einige Sätze über Kettenbrüche, mit Anwendungen auf die Theorie der Diophantischen Approximationen, Math. Ann., 92 (1924), 115-125. | JFM 50.0125.01

[KN] I. Kuipers, H. Niederreiter, Uniform distribution of sequences, Wiley, New-York-London, 1974. | Zbl 0281.10001

[L] S. Lang, Introduction to diophantine approximations, Addison-Wesley, Mass., 1966. | MR 35 #129 | Zbl 0144.04005

[P] O. Perron, Die Lehre von den Kettenbrüchen, 3 Aufl., Teubner, Stuttgart, 1954. | Zbl 0056.05901

[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 0161.04902

[R2] B. Randol, On the Fourier transform of the indicator function of a planar set, Trans. A.M.S., 139 (1969), 271-278. | MR 40 #4678a | Zbl 0183.26904

[Sch] W.M. Schmidt, Diophantine approximation, Lecture Notes in Math., 785, Springer-Verlag, Berlin, New York, 1980. | MR 81j:10038 | Zbl 0421.10019

[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 90g:11083 | Zbl 0693.41029

[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 0714.11045

[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).