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Integer points in the neighborhood of a plane curve of class \(C^n\). II. (Points entières au voisinage d’une courbe plane de classe \(C^n\). II.) (French. English summary) Zbl 1196.11136
Summary: M. Filaseta and O. Trifonov [Proc. Lond. Math. Soc. (3) 73, No. 2, 241–278 (1996; Zbl 0867.11053)] introduced a divisibility argument into the divided differences method for bounding the number of integer points in a narrow strip close to a curve; their result is useful when the strip is very narrow. We sharpen their argument, and extend it to allow broader strips. An immediate complication is the possible appearance of major arcs, which were first encountered in our earlier paper [Acta Arith. 69, No. 4, 359–366 (1995; Zbl 0822.11070)].

MSC:
11P21 Lattice points in specified regions
11J25 Diophantine inequalities
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Full Text: DOI Euclid
References:
[1] M. Branton et O. Ramaré, Nombre de racines d’un polynôme entier modulo \(q\), J. Théor. Nombres Bordeaux 10(1) (1998), 125–134. · Zbl 0916.11015
[2] M. Branton et P. Sargos, Points entiers au voisinage d’une courbe plane à très faible courbure. Bull. Sci. Math. 118 (1994), 15–28. · Zbl 0798.11039
[3] M. Filseta et O. Trifonov, The distribution of fractional parts with applications to gap results in number theory. Proc. London Math. Soc (3) 73 (1996), 241–278. · Zbl 0867.11053
[4] E. Fouvry et H. Iwaniec, Exponential sums for monomials. J. Number Theory 33 (1989), 311–333. · Zbl 0687.10028
[5] S.W. Graham et G. Kolesnik, Van der Corput’s method of exponential sums. Cambridge University Press (1991). · Zbl 0713.11001
[6] M.N. Huxley, The integer points close to a curve. Mathematika 36 (1989), 198–215. · Zbl 0659.10032
[7] M.N. Huxley, Exponential sums and lattice points. Proc. London Math. Soc. (3) 60 (1990), 471–502. · Zbl 0659.10057
[8] M.N. Huxley, Exponential sums and rounding error. J. London Math. Soc. (2) 43 (1991), 367–384. · Zbl 0687.65051
[9] M.N. Huxley et G. Kolesnik, Exponential sums with a large second derivative. Number Theory (ed. M. Jutila, et T. Metsänkylä) De Gruyter, Berlin (2001), 131–144. · Zbl 0966.11035
[10] M.N. Huxley et P. Sargos, Points entiers au voisinage d’une courbe plane de classe \(C^n\). Acta Arith. 69 (1995), 359–366. · Zbl 0822.11070
[11] P. Sargos, Points entiers au voisinage d’une courbe, sommes trigonométriques courtes et paires d’exposants. Proc. London Math. Soc. (3) 70 (1995), 285–312. · Zbl 0852.11042
[12] J. Stoer et R. Bulirsch, Introduction to numerical analysis. Springer Verlag (1980). · Zbl 0423.65002
[13] M. Branton et O. Ramaré, Nombre de racines d’un polynôme entier modulo \(q\), J. Théor. Nombres Bordeaux 10(1) (1998), 125–134. · Zbl 0916.11015
[14] M. Branton et P. Sargos, Points entiers au voisinage d’une courbe plane à très faible courbure. Bull. Sci. Math. 118 (1994), 15–28. · Zbl 0798.11039
[15] M. Filseta et O. Trifonov, The distribution of fractional parts with applications to gap results in number theory. Proc. London Math. Soc (3) 73 (1996), 241–278. · Zbl 0867.11053
[16] E. Fouvry et H. Iwaniec, Exponential sums for monomials. J. Number Theory 33 (1989), 311–333. · Zbl 0687.10028
[17] S.W. Graham et G. Kolesnik, Van der Corput’s method of exponential sums. Cambridge University Press (1991). · Zbl 0713.11001
[18] M.N. Huxley, The integer points close to a curve. Mathematika 36 (1989), 198–215. · Zbl 0659.10032
[19] M.N. Huxley, Exponential sums and lattice points. Proc. London Math. Soc. (3) 60 (1990), 471–502. · Zbl 0659.10057
[20] M.N. Huxley, Exponential sums and rounding error. J. London Math. Soc. (2) 43 (1991), 367–384. · Zbl 0687.65051
[21] M.N. Huxley et G. Kolesnik, Exponential sums with a large second derivative. Number Theory (ed. M. Jutila, et T. Metsänkylä) De Gruyter, Berlin (2001), 131–144. · Zbl 0966.11035
[22] M.N. Huxley et P. Sargos, Points entiers au voisinage d’une courbe plane de classe \(C^n\). Acta Arith. 69 (1995), 359–366. · Zbl 0822.11070
[23] P. Sargos, Points entiers au voisinage d’une courbe, sommes trigonométriques courtes et paires d’exposants. Proc. London Math. Soc. (3) 70 (1995), 285–312. · Zbl 0852.11042
[24] J. Stoer et R. Bulirsch, Introduction to numerical analysis. Springer Verlag (1980). · Zbl 0423.65002
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