On integer points in polygons

Maxim Skriganov

Annales de l'institut Fourier (1993)

  • Volume: 43, Issue: 2, page 313-323
  • ISSN: 0373-0956

Abstract

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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 ≪ ℓ n q t with some q > 0 , or as ≪ t ϵ with arbitrarily small ϵ > 0 .

How to cite

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Skriganov, Maxim. "On integer points in polygons." Annales de l'institut Fourier 43.2 (1993): 313-323. <http://eudml.org/doc/74997>.

@article{Skriganov1993,
abstract = {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\rightarrow \infty $, of certain classes of irrational polygons the error terms are bounded as $\ll \ell n^q t$ with some $q&gt;0$, or as $\ll t^\varepsilon $ with arbitrarily small $\varepsilon &gt;0$.},
author = {Skriganov, Maxim},
journal = {Annales de l'institut Fourier},
keywords = {anomaly small error terms; lattice point problem; irrational polygons},
language = {eng},
number = {2},
pages = {313-323},
publisher = {Association des Annales de l'Institut Fourier},
title = {On integer points in polygons},
url = {http://eudml.org/doc/74997},
volume = {43},
year = {1993},
}

TY - JOUR
AU - Skriganov, Maxim
TI - On integer points in polygons
JO - Annales de l'institut Fourier
PY - 1993
PB - Association des Annales de l'Institut Fourier
VL - 43
IS - 2
SP - 313
EP - 323
AB - 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\rightarrow \infty $, of certain classes of irrational polygons the error terms are bounded as $\ll \ell n^q t$ with some $q&gt;0$, or as $\ll t^\varepsilon $ with arbitrarily small $\varepsilon &gt;0$.
LA - eng
KW - anomaly small error terms; lattice point problem; irrational polygons
UR - http://eudml.org/doc/74997
ER -

References

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  1. [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. Zbl0409.58011
  2. [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. Zbl48.0197.07JFM48.0197.07
  3. [Kh] A. KHINCHIN, Einige Sätze über Kettenbrüche, mit Anwendungen auf die Theorie der Diophantischen Approximationen, Math. Ann., 92 (1924), 115-125. JFM50.0125.01
  4. [KN] I. KUIPERS, H. NIEDERREITER, Uniform distribution of sequences, Wiley, New-York-London, 1974. Zbl0281.10001
  5. [L] S. LANG, Introduction to diophantine approximations, Addison-Wesley, Mass., 1966. Zbl0144.04005MR35 #129
  6. [P] O. PERRON, Die Lehre von den Kettenbrüchen, 3 Aufl., Teubner, Stuttgart, 1954. Zbl0056.05901
  7. [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. Zbl0161.04902
  8. [R2] B. RANDOL, On the Fourier transform of the indicator function of a planar set, Trans. A.M.S., 139 (1969), 271-278. Zbl0183.26904MR40 #4678a
  9. [Sch] W.M. SCHMIDT, Diophantine approximation, Lecture Notes in Math., 785, Springer-Verlag, Berlin, New York, 1980. Zbl0421.10019MR81j:10038
  10. [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. Zbl0693.41029MR90g:11083
  11. [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. Zbl0714.11045
  12. [S3] M.M. SKRIGANOV, Construction of uniform distributions in terms of geometry of numbers, Prépublication de l'Institut Fourier, n° 200, Grenoble, 1992. 
  13. [S4] M.M. SKRIGANOV, Anomaly small errors in the lattice point problem, (in preparation). 

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