Let $H\subset \mathbb{R}^n$ with dimension ${\rm dim}(H)=\ell<n$; $S$ 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.

Let $H\subset \mathbb{R}^n$ with dimension ${\rm dim}(H)=\ell<n$; 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.