We derive a general upper bound for the number of incidences with $k$-dimensional varieties in ${\mathbb{R}}^d$. The leading term of this new bound generalizes previous bounds for the special cases of $k=1,k=d-1,$ and $k=d∕2$, to every $1\le k<d$. We also derive lower bounds showing that this leading term is tight in various cases, up to sub-polynomial factors.

To prove our incidence bounds, we introduce the dimension ratio of an incidence problem. This ratio provides an intuitive approach for deriving incidence bounds and isolating the main difficulties. It may also be a step towards a more unified incidence theory.

中文翻译：

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一般发生率 ${\mathbb{[R}}^d$

我们推导了一个事件发生次数的一般上限 $ķ$中的三维变体 ${\mathbb{[R}}^d$。此新界限的前导项概括了以下特殊情况的先前界限：$ķ=\mathrm{1个}，ķ=d-\mathrm{1个}，$ 和 $ķ=d∕\mathrm{2个}$，对每个 $\mathrm{1个}\le ķ<d$。我们还得出下界，表明该主导项在各种情况下都是紧紧的，直到次多项式因子为止。

为了证明我们的发病率边界，我们介绍了一个发病率问题的尺寸比。该比率提供了一种直观的方法来推导入射范围并隔离主要困难。这也可能是朝着更统一的发病率理论迈出的一步。