Geodesic
On the Distribution of the Sequence n2θ (mod 1)
Canadian journal of mathematics, Tome 39 (1987) no. 2, pp. 338-344

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

In a paper with the above title H. Heilbronn [4] proved that, given any real θ, there exist infinitely many positive integers n such that the distance ║θn 2║ from θn 2 to its nearest integral neighbour satisfies the bound He actually proved a somewhat stronger statement which shows that the integers n occur with some regularity and he suggested that perhaps the exponent may be replaced by . This theorem has attracted considerable attention and spawned a number of generalizations (see [1], [7] and the references therein), yet no essential improvement has been given for the original problem (but see [6], [8]). 
Friedlander, J. B.; Iwaniec, H. On the Distribution of the Sequence n2θ (mod 1). Canadian journal of mathematics, Tome 39 (1987) no. 2, pp. 338-344. doi: 10.4153/CJM-1987-016-2
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[1] 1. Baker, R. C., Recent results on fractional parts of polynomials, Number Theory, Carbondale (1979), 10–18, Lecture Notes in Math. 751 (Springer, Berlin, 1979). Google Scholar

[2] 2. Bombieri, E. and Iwaniec, H., On the order of ζ(½ + it), Ann. Sc. Abrm. Sup. Pisa (to appear). Google Scholar

[3] 3. Friedlander, J. B. and Iwaniec, H., Incomplete Kloosterman sums and a divisor problem, Annals of Math. 121 (1985), 319–350. Google Scholar

[4] 4. Heilbronn, H., On the distribution of the sequence n2θ (mod 1), Quart. J. Math. (Oxford) 19 (1948), 249–256. Google Scholar

[5] 5. Hooley, C., On the greatest prime factor of a cubic polynomial, J. Reine Angew. Math. 303/304 (1918), 21–50. Google Scholar

[6] 6. Hooley, C., On a new technique and its applications to the theory of numbers, Proc. London Math. Soc. (3) 38 ( 1979), 115–151. Google Scholar

[7] 7. Schmidt, W. M., Small fractional parts of polynomials, Regional Conference Series in Math. 32 (A.M.S., Providence, 1977). Google Scholar

[8] 8. Tenenbaum, G., Sur la concentration moyenne des diviseurs, Comm. Math. Helv. 60 (1985), 411–428. Google Scholar

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