Let me first recall what is the so-called paving conjecture: for any $\epsilon >0$, there exists $r\in \mathbb N$ such that for any bounded operator $A$ on $\ell^2(\mathbb Z)$, there exists a partition $(\mathcal P_s)_{1\le s\le r}$ of $\mathbb Z$, with

$$ \max_{1\le s\le r}\Vert P_s(A-\text{diag}A)P_s\Vert_{\mathcal B(\ell^2(\mathbb Z))}\le \epsilon \Vert A-\text{diag}A\Vert_{\mathcal B(\ell^2(\mathbb Z))}\quad\tag {PC} $$ where $P_s=\sum_{j\in \mathcal P_s} p_j$ and $p_j$ is the orthogonal projection onto $e_j=(\delta_{j,k})_{k\in \mathbb Z}$. The validity of that conjecture is still an open problem. It was also proven that the paving conjecture is equivalent to other famous conjectures, such as the Kadison-Singer conjecture and the Feichtinger conjecture. It is of course of key importance to notice that $r$ should depend only on $\epsilon$, that the partition could depend on the operator $A$, but not the number $r$ of elements in the partition.

Now my question: consider a Toeplitz matrix $T=(\phi(j-k))_{j,k\in \mathbb Z}$ giving rise to a bounded operator on $\ell^2(\mathbb Z)$. It is not known if $T$ is pavable in the sense that $(PC)$ holds for that class of matrices and I do not ask this question, probably as difficult as the general conjecture. Nevertheless, the Toeplitz matrices $T$ such that $\phi=\hat f$ with $f$ 1-periodic and Riemann integrable are known to be pavable (Halpern, Kaftal & Weiss). My question is: for a fixed $\epsilon$, say $\epsilon=1/2$, what is the dependence on $f$ of the number $r$ above? Looking at the proof, it seems that $r$ is not uniform for all Riemann integrable $f$. But on which type of quantitative behaviour of $f$ does $r$ depend?