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Small fractional parts of additive forms

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Abstract

Letf(x)=θ1 x k1 +...+θ s x k s be an additive form with real coefficients, and ∥α∥ = min {¦α-u¦:uεℤ} denote the distance fromα to the nearest integer. We show that ifθ 1,…,θ s , are algebraic ands = 4k then there are integersx 1,…,x s , satisfying l ≤x 1,≤ N and ∥f(x)∥ ≤ N E, withE = − 1 + 2/e.

Whens = λk, 1 ≤λ ≤ 2k, the exponentE may be replaced byλE/4, and if we drop the condition thatθ 1,…,θ s , be algebraic then the result holds for almost all values of θεℝs. Whenk ≥ 6 is small a better exponent is obtained using Heath-Brown’s version of Weyl’s estimate.

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  1. Department of Pure Mathematics, University of Sheffield, Hicks Building, Hounsfield Road, S3 7RH, Sheffield, England

    R. J. Cook

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  1. R. J. Cook
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Cook, R.J. Small fractional parts of additive forms. Proc. Indian Acad. Sci. (Math. Sci.) 99, 147–153 (1989). https://doi.org/10.1007/BF02837801

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