# zbMATH — the first resource for mathematics

On integer points in polygons. (English) Zbl 0779.11041
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, $$t\to\infty$$, of certain classes of irrational polygons the error terms are bounded as $$\ll\ln^ q t$$ with some $$q>0$$, or as $$\ll t^ \varepsilon$$ with arbitrarily small $$\varepsilon>0$$.

##### MSC:
 11P21 Lattice points in specified regions 11K38 Irregularities of distribution, discrepancy 11H06 Lattices and convex bodies (number-theoretic aspects)
Full Text:
##### References:
 [1] 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. · Zbl 0409.58011 [2] 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 [3] A. KHINCHIN, Einige Sätze über kettenbrüche, mit Anwendungen auf die Theorie der Diophantischen Approximationen, Math. Ann., 92 (1924), 115-125. · JFM 50.0125.01 [4] I. KUIPERS, H. NIEDERREITER, Uniform distribution of sequences, Wiley, New-York-London, 1974. · Zbl 0281.10001 [5] S. LANG, Introduction to Diophantine approximations, Addison-Wesley, Mass., 1966. · Zbl 0144.04005 [6] O. PERRON, Die lehre von den kettenbrüchen, 3 Aufl., Teubner, Stuttgart, 1954. · Zbl 0056.05901 [7] 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 [8] B. RANDOL, On the Fourier transform of the indicator function of a planar set, Trans. A.M.S., 139 (1969), 271-278. · Zbl 0183.26904 [9] W.M. SCHMIDT, Diophantine approximation, Lecture Notes in Math., 785, Springer-Verlag, Berlin, New York, 1980. · Zbl 0421.10019 [10] M.M. SKRIGANOV, On lattices in algebraic number fields, Dokl. Akad. Nauk SSSR, 306 (1989), 553-555, Soviet Math. Dokl., 39 (1989), 538-540. · Zbl 0693.41029 [11] 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 [12] M.M. SKRIGANOV, Construction of uniform distributions in terms of geometry of numbers, Prépublication de l’Institut Fourier, n° 200, Grenoble, 1992. [13] M.M. SKRIGANOV, Anomaly small errors in the lattice point problem, (in preparation). · Zbl 0976.11046
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.