Waring's problem with shifts

الأصل · EN

Let μ₁,, μₛ be real numbers, with μ₁ irrational. We investigate sums of shifted kth powers F(x₁,, xₛ) = (x₁ - μ₁)ᵏ + + (xₛ - μₛ)ᵏ. For k ≥ 4, we bound the number of variables needed to ensure that if η is real and τ> 0 is sufficiently large then there exist integers x₁ > μ₁,, xₛ > μₛ such that |F(x) - τ| < η. This is a real analogue to Waring's problem. When s ≥ 2k²-2k+3, we provide an asymptotic formula. We prove similar results for sums of general univariate degree k polynomials.