Geodesic
Probability and ergodic laws in the distribution of the~fractional parts of the~values of polynomials
Sbornik. Mathematics, Tome 192 (2001) no. 2, pp. 261-276

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In the present paper the fractional parts of the values of a polynomial are regarded as random variables depending on a randomly chosen vector whose coordinates are all the coefficients of the polynomial except the leading coefficient, which is assumed to be fixed. It is proved that the fractional parts and the distances between them are equally distributed and independent; the strong and superstrong laws of large numbers and the central limit theorem are proved for them. The probability distribution is found for the fractional part of a sum of the values of polynomials, which turns out to be universal. 
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     author = {L. D. Pustyl'nikov},
     title = {Probability and ergodic laws in the distribution of the~fractional parts of the~values of polynomials},
     journal = {Sbornik. Mathematics},
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L. D. Pustyl'nikov. Probability and ergodic laws in the distribution of the~fractional parts of the~values of polynomials. Sbornik. Mathematics, Tome 192 (2001) no. 2, pp. 261-276. http://geodesic.mathdoc.fr/item/SM_2001_192_2_a5/