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\begin{document}

\title[Countable-dimensional spaces of Fourier quasicrystals]{Countably infinite-dimensional spaces of Fourier quasicrystals and crystalline measures}

\author{Jamie Martin}
% Author: add affiliation and e-mail address with \address and \email.
\date{8 October 2026}

\subjclass[2020]{Primary 42A38; Secondary 46F12, 52C23, 42A75}
\keywords{crystalline measure, Fourier quasicrystal, Poisson summation
formula, Fourier uniqueness pair, Hermite expansion, Hamel dimension, Lee--Yang
polynomial}

\begin{abstract}
Meyer asked what the dimension of the space of crystalline measures with support and
spectrum in a given locally finite set $\Lambda\subset\R$ can be. We construct a set
$\Lambda$ on which the Fourier quasicrystals with support and spectrum in $\Lambda$
form a space of countably infinite Hamel dimension: every such Fourier quasicrystal
is a finite combination of a fixed sequence, and the layers of fixed polynomial
growth are finite-dimensional, with unbounded dimensions. This determines the
Fourier quasicrystals on $\Lambda$, not all of its crystalline measures. The blocks
of the construction are Poisson formulas on zero sets of Lee--Yang exponential
polynomials, with masses that grow polynomially at near-crossings of their sheets;
every finite union of blocks is determined by a uniqueness theorem for measures on
cones and an algebraic splitting on $\mathbb P^1\times\mathbb P^1$, and the infinite
union by compactness. We also construct a symmetric locally finite set $\Lambda$ for
which the space $\M_\Lambda$ of all crystalline measures has dimension $\aleph_0$, with
finite-dimensional layers of fixed Hermite--Sobolev order. Its blocks use clusters
of atoms that imitate derivatives of Dirac masses, placed at random scales, and no
nonzero element is a Fourier quasicrystal. The existence of sets with each of these
two properties (for the second, without the symmetry) has been formalised and
machine-checked in Lean~4, as recorded in Palomar \cite{PalStr,Pal26}.
\end{abstract}

\maketitle

\input{sec1_intro}
\input{sec2_literature}
\input{sec3_prelim}
\input{sec8_fqcount}
\input{sec4_blocks}
\input{sec5_exhaustion}
\input{sec6_pairs}
\input{sec7_fq}
\input{sec9_questions}

\appendix
\input{appD_fq}
\input{appA_rapid}
\input{appB_lift}
\input{appC_sec5}
\input{appE_lean}

\section*{Provenance and use of AI tools}
% Author: adapt this statement to the policy of the target journal.
The result on all crystalline measures (Theorem~\ref{thm:A}, Part~II) was obtained
by GPT-5.6 Sol, working in Codex, on 15 September 2026. It was found mostly
autonomously, using a custom auto-research harness that the author developed to make
autonomous discovery more efficient. The Lean formalisation of its existence
assertion (Theorem~\ref{thm:A}(i), without the symmetry of $\Lambda$) was updated five
days later, on 20 September 2026, and was submitted to Palomar on 30 September 2026
\cite{Pal26}. The results on Fourier quasicrystals (Part~I, the main result,
Section~\ref{sec:fqcount}) were obtained with
the same harness on 3 October 2026, and the formalisation of the existence statement
of Theorem~\ref{thm:D} was registered in Palomar on 8 October 2026 \cite{PalStr}.
The paper was written by Claude Opus 5.5.

The author is not a professional mathematician. The author has reviewed the paper
as carefully as possible but, as a non-specialist in this area, has limited
ability to check it beyond the verification tooling described in
Appendix~\ref{app:lean}. The author takes full responsibility for its content.
% \section*{Acknowledgements} To be completed by the author.

\input{refs}

\end{document}