# Small fractional parts of additive forms

• R. J. Cook
Article

## Abstract

Letf(x)=θ1 x 1 k +...+θ s x s k 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 θεℝ s . Whenk ≥ 6 is small a better exponent is obtained using Heath-Brown’s version of Weyl’s estimate.

## Keywords

Additive forms Heath-Brown’s version Weyl’s estimate fractional parts

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