Constructive syndetic partitions via restricted DFT matroids and irrational rotations

Definitions and mathematical statement

The workshop note asks how to construct a partition of exponential frames into "synthetic sets," referring to Bourgain--Tzafriri and the MSS existence result. Its report interprets the relevant target as syndetic Riesz classes; that terminology repair is not verified by a definition in the original. The appended "Bill Johnson" is retained as source text without assuming it supplies an unambiguous attribution.

In the report's model let $S\subset\mathbb T={\mathbb{R}}/{\mathbb{Z}}$ have normalized measure $m>0$. An exponential set indexed by $\Lambda\subset{\mathbb{Z}}$ is a Riesz sequence in $L^2(S)$ if some $0<A\leq B<\infty$ satisfy \[ A\sum|c_k|^2\leq\int_S\left|\sum_{k\in\Lambda}c_ke^{2\pi ikt}\right|^2dt \leq B\sum|c_k|^2 \] for every finitely supported coefficient family. Syndetic means bounded gaps in ${\mathbb{Z}}$. The question is to construct a full finite partition of the integer frequency set into such classes, with a specified effective input model for $S$.

The candidate rational-grid synthesis takes \[ S_A=\bigcup_{a\in A}[a/N,(a+1)/N),\quad \varnothing\ne A\subseteq\{0,\ldots,N-1\},\quad s=|A|. \] Using the $s$-by-$N$ restricted unnormalized DFT matrix $V=(e^{2\pi iab/N})_{a\in A,\,0\leq b<N}$, the report asserts that its vector matroid has arboricity $\lceil N/s\rceil$. Edmonds matroid partition then divides the columns into that many linearly independent sets $B_j$, and \[ \Lambda_j=B_j+N{\mathbb{Z}} \] gives periodic syndetic Riesz classes. The exact Riesz bounds are the least and greatest squared singular values of $V_{A,B_j}$, divided by $N$. The report claims the number $\lceil N/s\rceil$ is globally optimal, not merely optimal among the displayed periodic constructions.

For every open $S$ of measure $m$, the report further claims an equal-window irrational-rotation coloring with \[ r=\lfloor1/m\rfloor+1 \] Riesz classes, each with gaps at most $2r$. In such a coloring the classes are rotation itineraries of equal windows; the summary does not specify the required irrational rotation or the quantitative Riesz bounds, and does not claim every irrational rotation works.

Finally, a consequence attributed to Londner--Olevskii says some measurable spectra admit no nonempty periodic Riesz set. Thus periodic grid constructions do not solve the arbitrary-measurable constructive problem, which the report leaves unresolved. The input/representation meaning of constructive for a general measurable spectrum is not supplied.

Short English statement

The report interprets the note as constructive syndetic Riesz partitions. It gives an optimal restricted-DFT matroid construction for rational-grid spectra and an open-spectrum rotation construction, while retaining a periodicity obstruction for general measurable spectra.

Sources

• AIM workshop "Beyond Kadison--Singer: paving and consequences," item b15: https://docs.google.com/document/d/1XRw0WyG5PshqboCp99tovCRD1GCK0jXZiNLH_nw618c/edit. The source document was not accessed here.
• Local canonical registry aim-analysis-notes.json, notes[87], tag question; permanent UnsolvedMath ID $20000556$, source code catalogue entry. Original says "synthetic sets," names Bourgain--Tzafriri and MSS, asks for a constructive method, and appends Bill Johnson. Original remarks and literature field are empty.
• Supplied research attempt 1 invokes Edmonds matroid partition and a Londner--Olevskii consequence and proposes the combined constructions. No full authoritative references, rotation parameters, or arbitrary-measurable algorithm are supplied. The report is machine generated, not peer reviewed, and requires independent verification before citation.