Motivation and Mathematical Background

A natural starting point for the series is a very classical functional equation
arising in wavelet theory. Consider, for example, the indicator function
\(\varphi(x) = \mathbf{1}_{[0,1)}(x)\). A short calculation shows that \(\varphi\) satisfies the
scaling equation

\[
\varphi(x) = \varphi(2x) + \varphi(2x-1),
\]

where \(\varphi(2x)\) lives on \([0,\tfrac12)\) and \(\varphi(2x-1)\) on \([\tfrac12,1)\). Geometrically,
the function is recovered by dilating by a factor of two and then translating one of the pieces. Functions
that solve such a refinement equation are called scaling functions, and they
lie at the foundation of multiresolution analysis and the construction of modern wavelets.

In multiresolution analysis, one typically begins with a scaling function \(\varphi\) satisfying a more
general relation
\[
\varphi(x) = \sum_{k\in\mathbb{Z}} h_k \,\varphi(2x-k),
\]
and then constructs an associated wavelet \(\psi\). This explains why mathematicians were led to questions
about existence and structure of solutions to such functional equations: they are not toy problems, but
the first step in building constructions used every day in signal processing.

From Wavelets to the HRT Conjecture

Time–frequency analysis replaces pure dilations and translations by more flexible operators. For
\((x,y)\in\mathbb{R}^2\) and \(f\in L^2(\mathbb{R})\), we define the translation and modulation operators

\[
(T_x f)(t) = f(t-x), \qquad (M_y f)(t) = e^{2\pi i y t} f(t),
\]

and the corresponding time–frequency shift
\(\pi(x,y)f = M_yT_x f\). A finite Gabor system is then a collection
\(\{\pi(x_k,y_k)f : 1\le k\le n\}\) obtained from a single nonzero \(f\) and finitely many points
\((x_k,y_k)\) in the time–frequency plane.

The HRT Conjecture asks whether these finite Gabor systems are always linearly independent. More
precisely, it states that if

\[
\sum_{k=1}^n c_k\,\pi(x_k,y_k)f = 0 \quad\text{in }L^2(\mathbb{R}),
\]

with \(f\neq 0\) and distinct \((x_k,y_k)\), then all coefficients \(c_k\) must vanish. This deceptively
simple question remains wide open in full generality and has motivated a large body of work at the
interface of harmonic analysis, operator theory, and dynamical systems.

The Time–Frequency Plane

It is convenient to visualize these operators on the time–frequency plane
\(\mathbb{R}^2\): the horizontal axis represents time and the vertical axis frequency. Each point
\((x,y)\) labels a shift \(\pi(x,y)\), and a finite configuration of points corresponds to a finite Gabor
system. Special patterns (such as points on a line, a lattice, or a lattice plus one “outlier”) lead to
different versions and partial results around HRT.

Fourier and Zak Transforms as Main Tools

A first warm-up is the case of pure translations. Suppose we have distinct real numbers \(x_1,\dots,x_n\)
and a nonzero function \(f\in L^2(\mathbb{R})\), and that

\[
\sum_{k=1}^n c_k\,f(t-x_k) = 0 \quad\text{for a.e. }t\in\mathbb{R}.
\]

Taking Fourier transforms (initially on \(L^1\cap L^2\) and extending by continuity), translations in time
become modulations in frequency:

\[
0 = \sum_{k=1}^n c_k\,e^{-2\pi i x_k\xi}\,\widehat{f}(\xi)
= P(\xi)\,\widehat{f}(\xi),
\]

where \(P(\xi)=\sum_{k=1}^n c_k e^{-2\pi i x_k\xi}\) is a trigonometric polynomial. If not all \(c_k\) are
zero, then \(P\) is nontrivial and its zero set has Lebesgue measure zero. On the other hand,
\(\widehat{f}\) is nonzero on a set of positive measure. Hence the product \(P(\xi)\widehat{f}(\xi)\) cannot
vanish almost everywhere unless \(P\equiv 0\), and we conclude that all \(c_k = 0\). This illustrates the
power of transform methods for proving linear independence of structured systems.

For more complicated configurations, the Zak transform enters. For a suitable
\(f\in L^2(\mathbb{R})\), the Zak transform is defined by

\[
Zf(t,\omega) = \sum_{j\in\mathbb{Z}} f(t+j) e^{-2\pi i j\omega},
\]

and extends to a unitary operator
\(Z : L^2(\mathbb{R}) \to L^2\big([0,1)^2\big)\). It is periodic in \(\omega\), quasi-periodic in \(t\), and
interacts particularly well with lattice shifts:

\[
Z\big(M_y T_x f\big)(t,\omega) = e^{-2\pi i(yt + x\omega)}\,Zf(t,\omega), \quad x,y\in\mathbb{Z}.
\]

As a consequence, if a finite linear combination of lattice shifts vanishes, the Zak transform converts it
into a trigonometric polynomial in two variables times \(Zf\). Arguing as above with zero sets of
trigonometric polynomials and the unitarity of \(Z\), one proves HRT for all lattice configurations:
finitely many lattice time–frequency shifts of a nonzero \(f\) are linearly independent.

Mixed-Integer Configurations and Rigidity

The series then turns to mixed-integer configurations, where most points lie
on a lattice such as \(\mathbb{Z}^2\), but one additional point \((\alpha,\beta)\) does not. In the
language of the time–frequency plane, this is a “lattice plus one” configuration. The full technical
treatment of this case, including the three-orbit viewpoint underlying Lecture 2, is developed in the
preprint
available on arXiv.

The central idea is to start from a hypothetical linear dependence, translate it into a functional
equation for the Zak transform, and then iterate this equation along orbits generated by a vector
\(\gamma\) encoding the off-lattice point. The quasi-periodicity of \(Zf\) forces a strong
rigidity: under suitable arithmetic hypotheses on \(1,\alpha,\beta\), the
only way for such a functional equation to hold is for \(Zf\) (and thus \(f\)) to be identically zero.
This contradicts the assumption that \(f\neq 0\), and so no nontrivial linear dependence can exist for
that configuration.

In the case where \(1,\alpha,\beta\) are rationally independent, the orbit of a point in the torus
\([0,1)^2\) under translation by \(\gamma\) is dense. Combined with continuity of \(Zf\) (for instance,
if \(f\) is a Schwartz function), this implies that a single zero forces many more zeros, and, via
density, global vanishing. This mechanism is one of the main tools in the “mixed-integer” part of
the story.

Ergodicity and Atkinson’s Lemma

The final lecture brings in a more dynamical viewpoint. One associates to the phase factors appearing in
the Zak transform a compact abelian group \(H\) and a measure-preserving transformation
\(T : H \to H\) given by multiplication by a fixed element. A logarithm of the modulus of a certain
trigonometric polynomial becomes a function \(\Phi \in L^1(H)\) with mean zero.

Atkinson’s lemma, a refinement of Birkhoff’s ergodic theorem, then says that for almost every point
\(h\in H\) there is a subsequence \(n_r \to \infty\) along which partial sums

\[
\sum_{j=0}^{n_r-1} \Phi(T^j h)
\]

exhibit a form of recurrence. In the HRT setting, these sums control the growth or decay of the Fourier
transform \(\widehat{f}\) along certain arithmetic progressions. Combining Atkinson’s lemma with the fact
that \(\widehat{f}(\xi)\to 0\) as \(|\xi|\to\infty\) for \(f\in L^1\cap L^2\) allows one to rule out linear
dependencies for additional configurations, including a “one-line-plus-one” configuration where most points
lie on a single line and one point is off it.

The technical heart of this approach is to reconcile “almost every” statements from ergodic theory with the
very specific points dictated by the time–frequency configuration. This leads naturally to questions about
subsequences, cluster points on compact groups, and the extent to which one can pass from generic orbits to
particular algebraic ones.

Outlook

Taken together, these lectures show how ideas from classical harmonic analysis (Fourier and Zak transforms),
wavelet theory (scaling equations and multiresolution analysis), and ergodic theory (Atkinson-type
recurrence on compact groups) fit together in the ongoing effort to understand the HRT Conjecture. Even in
the restricted settings where we now have complete results—lattices, mixed-integer configurations under
arithmetic constraints, or one-line-plus-one patterns—the interplay between algebra, geometry, and dynamics
is surprisingly rich and suggests new directions for both analysis and dynamical systems.

The seminar is designed so that a graduate student or an advanced undergraduate with a background in real
analysis and basic Fourier theory can follow the main ideas, while still pointing toward open problems and
current research directions in time–frequency analysis.