A new lower bound for (3/2) k
Journal de théorie des nombres de Bordeaux, Tome 19 (2007) no. 1, pp. 311-323.

Nous démontrons que pour tout entier k supérieur à une constante K effectivement calculable, la distance de (3/2) k à l’entier le plus proche est minorée par 0,5803 k .

We prove that, for all integers k exceeding some effectively computable number K, the distance from (3/2) k to the nearest integer is greater than 0.5803 k .

DOI : 10.5802/jtnb.588
Zudilin, Wadim 1

1 Department of Mechanics and Mathematics Moscow Lomonosov State University Vorobiovy Gory, GSP-2 119992 Moscow, Russia
@article{JTNB_2007__19_1_311_0,
     author = {Zudilin, Wadim},
     title = {A new lower bound for ${\Vert (3/2)^k\Vert }$},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {311--323},
     publisher = {Universit\'e Bordeaux 1},
     volume = {19},
     number = {1},
     year = {2007},
     doi = {10.5802/jtnb.588},
     zbl = {1127.11049},
     mrnumber = {2332068},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/jtnb.588/}
}
TY  - JOUR
AU  - Zudilin, Wadim
TI  - A new lower bound for ${\Vert (3/2)^k\Vert }$
JO  - Journal de théorie des nombres de Bordeaux
PY  - 2007
SP  - 311
EP  - 323
VL  - 19
IS  - 1
PB  - Université Bordeaux 1
UR  - http://www.numdam.org/articles/10.5802/jtnb.588/
DO  - 10.5802/jtnb.588
LA  - en
ID  - JTNB_2007__19_1_311_0
ER  - 
%0 Journal Article
%A Zudilin, Wadim
%T A new lower bound for ${\Vert (3/2)^k\Vert }$
%J Journal de théorie des nombres de Bordeaux
%D 2007
%P 311-323
%V 19
%N 1
%I Université Bordeaux 1
%U http://www.numdam.org/articles/10.5802/jtnb.588/
%R 10.5802/jtnb.588
%G en
%F JTNB_2007__19_1_311_0
Zudilin, Wadim. A new lower bound for ${\Vert (3/2)^k\Vert }$. Journal de théorie des nombres de Bordeaux, Tome 19 (2007) no. 1, pp. 311-323. doi : 10.5802/jtnb.588. http://www.numdam.org/articles/10.5802/jtnb.588/

[1] A. Baker, J. Coates, Fractional parts of powers of rationals. Math. Proc. Cambridge Philos. Soc. 77 (1975), 269–279. | MR | Zbl

[2] M. A. Bennett, Fractional parts of powers of rational numbers. Math. Proc. Cambridge Philos. Soc. 114 (1993), 191–201. | MR | Zbl

[3] M. A. Bennett, An ideal Waring problem with restricted summands. Acta Arith. 66 (1994), 125–132. | MR | Zbl

[4] F. Beukers, Fractional parts of powers of rationals. Math. Proc. Cambridge Philos. Soc. 90 (1981), 13–20. | MR | Zbl

[5] G. V. Chudnovsky, Padé approximations to the generalized hypergeometric functions. I. J. Math. Pures Appl. (9) 58 (1979), 445–476. | MR | Zbl

[6] F. Delmer, J.-M. Deshouillers, The computation of g(k) in Waring’s problem. Math. Comp. 54 (1990), 885–893. | Zbl

[7] A. K. Dubickas, A lower bound for the quantity (3/2) k . Russian Math. Surveys 45 (1990), 163–164. | MR | Zbl

[8] L. Habsieger, Explicit lower bounds for (3/2) k . Acta Arith. 106 (2003), 299–309. | MR | Zbl

[9] J. Kubina, M. Wunderlich, Extending Waring’s conjecture up to 471600000. Math. Comp. 55 (1990), 815–820. | Zbl

[10] K. Mahler, On the fractional parts of powers of real numbers. Mathematika 4 (1957), 122–124. | MR | Zbl

[11] L. J. Slater, Generalized hypergeometric functions. Cambridge University Press, 1966. | MR | Zbl

[12] R. C. Vaughan, The Hardy–Littlewood method. Cambridge Tracts in Mathematics 125, Cambridge University Press, 1997. | MR | Zbl

[13] W. Zudilin, Ramanujan-type formulae and irrationality measures of certain multiples of π. Mat. Sb. 196:7 (2005), 51–66. | MR | Zbl

Cité par Sources :