Geodesic
On a Theorem of Heilbronn Concerning the Fractional Part of θn 2
Canadian journal of mathematics, Tome 22 (1970) no. 4, pp. 784-788

Voir la notice de l'article provenant de la source Cambridge University Press

1. In 1948 Heilbronn [4] proved the following theorem.THEOREM H. For every real θ and every positive integer N, there is an integer n satisfying (1.1) whereis an arbitrarily small number,depends only on, and ‖t‖ means the distance from t to the nearest integer.The interest of the result (1.1) is that the inequality is uniform in θ, and is therefore analogous to the classical inequality of Dirichlet for the fractional part of θn . In this paper we shall prove the following theorem.THEOREM. For every real θ and every positive integer N, there is an integer n satisfying (1.2) where A is an absolute constant and. 
Liu, Ming-Chit. On a Theorem of Heilbronn Concerning the Fractional Part of θn 2. Canadian journal of mathematics, Tome 22 (1970) no. 4, pp. 784-788. doi: 10.4153/CJM-1970-088-9
@article{10_4153_CJM_1970_088_9,
     author = {Liu, Ming-Chit},
     title = {On a {Theorem} of {Heilbronn} {Concerning} the {Fractional} {Part} of \ensuremath{\theta}n 2},
     journal = {Canadian journal of mathematics},
     pages = {784--788},
     year = {1970},
     volume = {22},
     number = {4},
     doi = {10.4153/CJM-1970-088-9},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CJM-1970-088-9/}
}
TY  - JOUR
AU  - Liu, Ming-Chit
TI  - On a Theorem of Heilbronn Concerning the Fractional Part of θn 2
JO  - Canadian journal of mathematics
PY  - 1970
SP  - 784
EP  - 788
VL  - 22
IS  - 4
UR  - http://geodesic.mathdoc.fr/articles/10.4153/CJM-1970-088-9/
DO  - 10.4153/CJM-1970-088-9
ID  - 10_4153_CJM_1970_088_9
ER  - 
%0 Journal Article
%A Liu, Ming-Chit
%T On a Theorem of Heilbronn Concerning the Fractional Part of θn 2
%J Canadian journal of mathematics
%D 1970
%P 784-788
%V 22
%N 4
%U http://geodesic.mathdoc.fr/articles/10.4153/CJM-1970-088-9/
%R 10.4153/CJM-1970-088-9
%F 10_4153_CJM_1970_088_9

[1] 1. Ayoub, R., An introduction to the analytic theory of numbers, Mathematical Surveys, No. 10 (Amer. Math. Soc, Providence, R.I., 1963). Google Scholar

[2] 2. Davenport, H., On a theorem of Heilbronn, Quart. J. Math. Oxford Ser. 18 (1967), 339–344. Google Scholar

[3] 3. Hardy, G. H. and Wright, E. M., An introduction to the theory of numbers, 3rd ed. (Oxford, at the Clarendon Press, 1954). Google Scholar

[4] 4. Heilbronn, H., On the distribution of the sequence n20 (mod 1), Quart. J. Math. Oxford Ser. 19 (1948), 249–256. Google Scholar

[5] 5. Hua, L. K., Additive theory of prime numbers, Translations of Mathematical Monographs, Vol. 13 (Amer. Math. Soc, Providence, R.I., 1965). Google Scholar

[6] 6. Vinogradov, I. M., The method of trigonometric sums in the theory of numbers (Interscience, New York, 1954). Google Scholar

Cité par Sources :