Sums of cubes with shifts

Sam Chow

Research output: Contribution to journalArticle (Academic Journal)

4 Citations (Scopus)
148 Downloads (Pure)

Abstract

Let μ1,…,μs be real numbers, with μ1 irrational. We investigate sums of shifted cubes F(x1,…,xs) = (x1μ1)3 + ⋯ + (xsμs)3. We show that if η is real, τ>0 is sufficiently large, and s9, then there exist integers x1>μ1,…,xs>μs such that |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 s5. For s11, we provide an asymptotic formula. If s6, then F(Zs) is dense on the reals. Given nine variables, we can generalize this to sums of univariate cubic polynomials.
Original languageEnglish
Pages (from-to)343-366
Number of pages24
JournalJournal of the London Mathematical Society
Volume91
Issue number2
Early online date9 Jan 2015
DOIs
Publication statusPublished - Apr 2015

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