Exact finite Feichtinger partition count and a sharp ceiling obstruction

Definitions and mathematical statement

Let $0<\epsilon\leq1$. A Parseval frame $(f_i)_{i\in I}$ in a complex Hilbert space $H$, with finite or countable $I$, satisfies \[ \sum_{i\in I}|\langle h,f_i\rangle|^2=\|h\|^2\quad(h\in H). \] Impose $\|f_i\|\geq\epsilon$ for every $i$; $\epsilon$ bounds norms, not squared norms. A subsequence indexed by $J\subset I$ is Riesz basic if there are $0<A_J\leq B_J<\infty$ such that \[ A_J\sum_{i\in J}|c_i|^2\leq\left\|\sum_{i\in J}c_if_i\right\|^2 \leq B_J\sum_{i\in J}|c_i|^2 \] for all finitely supported coefficient families on $J$. The source asks for the least number of blocks in a partition of $I$ into Riesz basic subsequences, its dependence on $\epsilon$, and ways to exhibit or construct such partitions.

Define $N(\epsilon)$ as the least integer $r$ such that every frame in this finite-or-countable class admits a partition into at most $r$ Riesz basic subsequences. The unquantified definition allows the Riesz constants to depend on the frame and blocks. Define $N_{\mathrm{fin}}(\epsilon)$ similarly for finite frames only, without prescribing quantitative Riesz constants. The aid reports that the published Bownik--Casazza--Marcus--Speegle theorem gives \[ N(\epsilon)=\Theta(\epsilon^{-2})\quad(\epsilon\downarrow0). \] This is a supplied literature attribution, not a complete theorem citation. The separate claimed exact finite result and countable lower bound are \[ N_{\mathrm{fin}}(\epsilon)=\lceil\epsilon^{-2}\rceil, \qquad N(\epsilon)\geq\lceil\epsilon^{-2}\rceil \quad(0<\epsilon\leq1). \] The exact finite statement is weaker in its Riesz-constant requirements than a dimension-independent quantitative Riesz bound, and it does not assert equality for the countable case.

The supplied Fourier witness may be made explicit as follows. Put $r=\lceil\epsilon^{-2}\rceil$. If $r\geq2$, choose integers $1\leq n<m$ with $r-1<m/n\leq\epsilon^{-2}$. With $\zeta=e^{2\pi i/m}$, the columns \[ f_j=\frac1{\sqrt m}(1,\zeta^j,\ldots,\zeta^{(n-1)j})^{\mathsf T}\in{\mathbb{C}}^n, \qquad0\leq j<m, \] form a Parseval frame, have norm $\sqrt{n/m}\geq\epsilon$, and are full spark, meaning every $n$ columns are linearly independent. Since every independent block contains at most $n$ columns and $m>(r-1)n$, at least $r$ blocks are needed. For $\epsilon=1$, an orthonormal basis gives the lower bound one.

For the finite upper bound, each subfamily satisfies \[ \epsilon^2|J|\leq\sum_{i\in J}\|f_i\|^2 \leq\dim\operatorname{span}\{f_i:i\in J\}. \] The Rado--Horn criterion therefore yields a partition into $r=\lceil\epsilon^{-2}\rceil$ linearly independent blocks; finite independent blocks are Riesz basic with their own positive constants. The aid reports an exact-arithmetic polynomial-time construction by linear-matroid partition. Its computation claim is an arithmetic-model assertion, not a supplied finite-bit complexity theorem for arbitrary complex input. No enumeration of all admissible partitions or exact countable-case optimum is supplied.

Short English statement

The question asks how many Riesz subsequences are needed to partition a Parseval frame whose vector norms are at least $\epsilon$. The supplied literature report gives order $\epsilon^{-2}$ for countable frames. The separate finite result gives exactly $\lceil\epsilon^{-2}\rceil$ blocks when Riesz constants may depend on the finite frame, with Fourier lower examples and a matroid-partition construction.

Sources

• Local registry: permanent UnsolvedMath ID $20000553$, exact public code catalogue entry; aim-analysis-notes.json, notes[84].
• AIM workshop Beyond Kadison--Singer: paving and consequences, item b12: https://docs.google.com/document/d/1XRw0WyG5PshqboCp99tovCRD1GCK0jXZiNLH_nw618c/edit. The original text refers informally to Feichtinger's conjecture and says "Pete C." would show results "tomorrow," without a date or full attribution.
• Supplied research summary, "Exact finite Feichtinger partition count and a sharp ceiling obstruction," attempt 1. It attributes the asymptotic count to Bownik--Casazza--Marcus--Speegle, the finite upper criterion to Rado--Horn, and the construction to linear-matroid partition. No full theorem citations or publication details are supplied; the original literature and remarks fields are empty.