# Intersection of a lower dimensional space and a discrete set

Let $$H\subset \mathbb{R}^n$$ with dimension $${\rm dim}(H)=\ell; let $$S$$ be a finite subset of reals.

My question is the following. Is it correct to say, $${\rm card}(H \cap V)\leqslant |S|^\ell$$ where $$V$$ is $$n$$-dimensional vectors with entries from $$S$$, that is, $$V=\{v\in\mathbb{R}^n: v_i\in S,\forall i\}$$.

I know this is related to several old results by Andrew Odlyzko. My reasoning is that, if $${\rm dim}(H)=\ell$$, there exists $$\ell$$ determining coordinates $$n_1,\dots,n_\ell$$, such that once we fix $$v_{n_1},\dots,v_{n_\ell}$$, the rest is uniquely determined. There are precisely $$|S|^\ell$$ such vectors.

Remark: This is connected to the study of the singularity probability of random $$n\times n$$ binary matrices, initiated by Komlos, then enriched by the works of Komlos, Kahn-Komlos-Szemeredi, Tao-Vu, Bourgain et al, and finally, by Tikhomirov.

We show $$|H\cap V|\leq |S|^\ell$$. The proof is by induction on $$\ell\leq n$$ and allows $$H$$ to be an affine subspace.
If $$\ell=0$$ then the result is clear. So fix $$\ell\geq 1$$ and assume the result for $$\ell-1$$. Fix a finite set $$S\subset\mathbb{R}$$ and let $$V=S^n$$. Let $$H$$ be an affine subspace of $$\mathbb{R}^n$$ of dimension $$\ell$$.
For $$i\leq n$$ and $$r\in\mathbb{R}$$, let $$W^r_i\subseteq\mathbb{R}^n$$ be the $$(n-1)$$-dimensional affine space of vectors whose $$i^{\text{th}}$$ coordinate is $$r$$. There is some $$i\leq n$$ such that for all $$r\in\mathbb{R}$$, $$H$$ is not contained in $$W^r_i$$. (Otherwise, if for all $$i\leq n$$, there is some $$r_i$$ such that $$H\subseteq W^{r_i}_i$$, then $$H=\{(r_1,\ldots,r_n)\}$$, contradicting $$\ell\geq 1$$.)
Now, for a contradiction suppose $$|H\cap V|\geq |S|^\ell+1$$. For $$s\in S$$, let $$H_s=H\cap W^s_i$$. Then $$\{H_s\cap V\}_{s\in S}$$ is a partition of $$H\cap V$$. So there is some $$s\in S$$ such that $$|H_s\cap V|\geq |S|^{\ell-1}+1$$. Since $$H$$ is not contained in $$W^s_i$$, it follows that $$H_s$$ is an affine space of dimension $$\ell-1$$. This contradicts the induction hypothesis.