A Hierarchical Online-Compression Approach to Proportional Subsets of Bounded Orthonormal Systems

Let $(ϕ_i)_{i=1}^n$ be an orthonormal system on a probability space with $\|ϕ_i\|_\infty\leq L$. We prove that, for every $2\leq k<n$, there is a set $I$ of at least $n-k$ indices such that $$ \|f\|_2\leq CM_k\sqrt{\log(e+M_k)}\|f\|_1 $$ for every $f$ in the span of $(ϕ_i)_{i\in I}$, where $M_k=L\sq