论文标题
单位立方体子集中简单乘积的定量界限
Quantitative bounds for product of simplices in subsets of the unit cube
论文作者
论文摘要
对于每个$ 1 \ leq i \ le n $，令$ k_i \ geq 1 $，让$Δ_i$是一组非排分简单的顶点，$ k_i+1 $ 1 $ point in $ \ mathbb {r}^{r}^{k_i+1} $。如果$ a \ subseteq [0,1]^{k_1+1} \ times \ cdots \ times \ times [0,1]^{k_n+1} $是一套可测量的度量集，至少$δ$，我们显示有一个间隔$ i = i = i（δ__1，\ ldots $ ldots，uddots $ udtots，udectal，upent，upent，a） $ \ exp（-Δ^{ - c（δ_1，\ ldots，Δ_n）}）$，以至于对于I $中的每个$λ\，set $ a $ cum cum co $Δ这是Lyall和Magyar结果的定量改进。我们的证明依赖于谐波分析。证明中的主要成分是与与$ n $ - 零件$ n $ regartular超图相关的多线性单数积分类似的表格的取消估计。
For each $1\leq i \le n$, let $k_i\geq 1$ and let $Δ_i$ be a set of vertices of a non-degenerate simplex of $k_i+1$ points in $\mathbb{R}^{k_i+1}$. If $A\subseteq [0,1]^{k_1+1}\times \cdots \times [0,1]^{k_n+1}$ is a Lebesgue measurable set of measure at least $δ$, we show that there exists an interval $I=I(Δ_1,\ldots, Δ_n,A)$ of length at least $\exp(-δ^{-C(Δ_1,\ldots, Δ_n)})$ such that for each $λ\in I$, the set $A$ contains $Δ'_1\times \cdots \times Δ'_n$, where each $Δ_i'$ is an isometric copy of $λΔ_i$. This is a quantitative improvement of a result by Lyall and Magyar. Our proof relies on harmonic analysis. The main ingredient in the proof are cancellation estimates for forms similar to multilinear singular integrals associated with $n$-partite $n$-regular hypergraphs.
