### Abstract

Let

*μ*_{1},…,**μ**_{s}be real numbers, with*μ*_{1}irrational. We investigate sums of shifted cubes*(***F***x*_{1},…,*x*) = (_{s}*x*_{1}−*μ*_{1})^{3 }+ ⋯ + (*x*−_{s}*μ*)_{s}^{3}. We show that if*η*is real,*τ*>**0**is sufficiently large, and*s*⩾**9**, then there exist integers*x*_{1}>*μ*_{1},…,*x*>_{s}*μ*such that |_{s}*(***F****x**)−*τ*|<*η*. This is a real analogue to Waring's problem. We then prove a full density result of the same flavour for*s*⩾**5**. For*s*⩾**11**, we provide an asymptotic formula. If*s*⩾**6**, then*(***F****Z**^{s}) is dense on the reals. Given nine variables, we can generalize this to sums of univariate cubic polynomials.Original language | English |
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Pages (from-to) | 343-366 |

Number of pages | 24 |

Journal | Journal of the London Mathematical Society |

Volume | 91 |

Issue number | 2 |

Early online date | 9 Jan 2015 |

DOIs | |

Publication status | Published - Apr 2015 |

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## Cite this

Chow, S. (2015). Sums of cubes with shifts.

*Journal of the London Mathematical Society*,*91*(2), 343-366. https://doi.org/10.1112/jlms/jdu077