凸曲线上点的近最优三重加性能量界
Near optimal three-fold additive energy bound for points on convex curves
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中文总结 AI 辅助
针对凸曲线上有限点集，证明其三重加性能量的近最优上界，并将其应用于推导凸序列差集与和集大小的下界
中文摘要 AI 辅助
设X为实数集ℝ的有限子集，γ(t)=(t,f(t))且f严格凸，我们证明J₃(γ(X))=#{(x₁,…,x₆)∈X⁶:∑₁³γ(xᵢ)=∑₄⁶γ(xᵢ)}≪_ε|X|^{3+ε}；作为应用，对任意凸序列A⊂ℝ，证得|A-A|≫_ε|A|^{5/3-ε}，|A+A|≫_ε|A|^{8/5-ε}
英文摘要
Let $X\subset\mathbb{R}$ be finite and let $γ(t)=(t,f(t))$, where $f$ is strictly convex. We show that \[ J_3(γ(X)) =\#\{(x_1,\ldots,x_6)\in X^6:\sum_{i=1}^3γ(x_i)=\sum_{i=4}^6γ(x_i)\} \ll_ε|X|^{3+ε}. \] When specialized to the parabola, our result implies near-optimal estimates for the number of solutions to the diameter-free quadratic Vinogradov system. As a second application, we settle a conjecture from Krishnapur-Kurlberg-Wigman and Bombieri-Bourgain concerning lattice points on dilates of the unit circle. As a third application, we prove that $|A-A|\gg_ε|A|^{5/3-ε}$ and $|A+A|\gg_ε|A|^{8/5-ε}$ for any finite convex sequence $A\subset \mathbb{R}$.
