论文标题
线性地图家族的禁止交点问题
Forbidden intersection problems for families of linear maps
论文作者
论文摘要
我们研究了线性地图家族的Erdős-Sós禁止交点问题的类似物。如果$ v $和$ w $是同一字段上的向量空间，我们说线性$ \ Mathcal {f} $从$ v $到$ w $ is \ emph {$（t-1）$ - 相互交流}，如果任何两个线性$σ_1，n in \ in \ mathcal v} σ_1（v）=σ_2（v）\}）\ neq t-1 $。我们证明，如果$ n $取决于$ t $，则$ q $是任何主要功率，$ v $是$ n $二维矢量空间，超过$ \ mathbb {f} _q $，而$ \ \ \ \ m athcal {f} \ subset {subset subset \ subset \ subset \ textrm {gl}（gl}（v）$（v）$（v） $ | \ MATHCAL {F} | \ leq \ prod_ {i = 1}^{n-t}（q^n-q^{i+t-1}）$。仅在存在$ \ nathcal {f} $的所有元素的$ v $的$ t $二维子空间时，平等才能保持，或$ v^*$的a $ t $二维子空间上的所有元素$ \ \ \ \ \ {σ^*：\ {我们的主要工具是线性地图的家庭的“ junta近似”结果，并具有禁止的交叉点：如果$ v $和$ w $是在同一有限字段上的有限维矢量空间，那么任何$（t-1）$ - 任何$（t-1）$ - 从$ v $ w $ w $ n $ w $ n $ w $ n junts a junt in j in a j in （意思是，一个家庭$ \ Mathcal {J} $的线性地图从$ V $到$ W $ $ v_1，\ ldots，v_m \ in v $，$ a_1，\ ldots，a_n \ in w^*$和$ m+n $的边界）。这反过来又依赖于“ junta方法”的变体（最初由Dinur和Friedgut引入，并由凯勒和最后一位作者延伸，以及光谱技术和超平等性。
We study an analogue of the Erdős-Sós forbidden intersection problem, for families of linear maps. If $V$ and $W$ are vector spaces over the same field, we say a family $\mathcal{F}$ of linear maps from $V$ to $W$ is \emph{$(t-1)$-intersection-free} if for any two linear maps $σ_1,σ_2 \in \mathcal{F}$, $\dim(\{v \in V:\ σ_1(v)=σ_2(v)\}) \neq t-1$. We prove that if $n$ is sufficiently large depending on $t$, $q$ is any prime power, $V$ is an $n$-dimensional vector space over $\mathbb{F}_q$, and $\mathcal{F} \subset \textrm{GL}(V)$ is $(t-1)$-intersection-free, then $|\mathcal{F}| \leq \prod_{i=1}^{n-t}(q^n - q^{i+t-1})$. Equality holds only if there exists a $t$-dimensional subspace of $V$ on which all elements of $\mathcal{F}$ agree, or a $t$-dimensional subspace of $V^*$ on which all elements of $\{σ^*:\ σ\in \mathcal{F}\}$ agree. Our main tool is a `junta approximation' result for families of linear maps with a forbidden intersection: namely, that if $V$ and $W$ are finite-dimensional vector spaces over the same finite field, then any $(t-1)$-intersection-free family of linear maps from $V$ to $W$ is essentially contained in a $t$-intersecting \emph{junta} (meaning, a family $\mathcal{J}$ of linear maps from $V$ to $W$ such that the membership of $σ$ in $\mathcal{J}$ is determined by $σ(v_1),\ldots,σ(v_M),σ^*(a_1),\ldots,σ^*(a_N)$, where $v_1,\ldots,v_M \in V$, $a_1,\ldots,a_N \in W^*$ and $M+N$ is bounded). The proof of this in turn relies on a variant of the `junta method' (originally introduced by Dinur and Friedgut, and powefully extended by Keller and the last author), together with spectral techniques and a hypercontractive inequality.
