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A partial digestion of the HRT counterexample

6 August, 2026 in expository, math.CA | Tags: HRT conjecture, time-frequency analysis | by Terence Tao

A function of one variable can be translated in space by a spatial shift to obtain a new function

and also modulated in frequency by a frequency shift to obtain a new function

One can compose these two operations to obtain a time-frequency shift:

(This can be viewed of as a portion of the Weyl representation of the Heisenberg group, but we will not adopt a representation-theoretic perspective here.)

Some functions obey finite linear relations between their time-frequency shifts. For instance, a sinusoid obeys the relation

However, the Heil-Ramanathan-Topiwala (HRT) conjecture states that once one imposes some reasonable decay condition on , no such relations exist:

Conjecture 1 (HRT conjecture) If is non-zero, then there is no relation of the form

for some distinct points and some coefficients , not all zero.

A special case of the HRT conjecture, which was also open, makes the additional assumption that was Schwartz.

Many positive results towards this conjecture were known. I will mention only a few here. There is a result of Linnell that the conjecture is true if lie in a translate of a discrete subgroup of ; this (together with an argument handling the collinear case) establishes all cases where , and several partial results involving the cases are also known. The conjecture is also known if is decays at a suitably super-exponential rate, by work of Bownik and Speegle.

I was aware of this conjecture through various talks and conversations with colleagues, and even briefly tried my hand at it for a while, though not with particularly serious effort (or progress). It was thus a nice surprise to see that it has just been resolved by Faulhuber, Petersen, van Velthoven, and Voigtlaender, even in the Schwartz case:

Theorem 2 There exist complex numbers , not all zero, distinct points , and a non-zero Schwartz function such that

It is perhaps unsurprising in this current era that this result is AI-assisted. However, I think the authors have disclosed their AI use responsibly, with the final arguments written by hand with a readable overview of the argument, as well as proper discussion of methods, relation to past literature, and other independent numerical checks on the result.

The negative result lies only a little beyond the positive results: is now increased to , and all but one of the points lie in (a translate of) a discrete subgroup of (in fact the explicit subgroup is used). The functions constructed are smooth and rapidly decaying, but not analytic or super-exponentially decaying, which would start being in conflict with the known positive results.

In addition to AI being used to come up with the initial proof strategy, a more traditional numerical computation was used to verify one step of the argument.

I have not had the time to do a full digestion of the result, but (after reading the introduction, and using a little AI assistance of my own) I was able to understand the main ideas at a high level. The first few reductions are relatively standard. Setting and , one can view the problem as one of solving an eigenvalue problem

The time-frequency shifts are chosen to lie in a translate of the discrete subgroup by a certain irrational shift . If one was working in the shifts of a standard lattice , it would be natural to work with the Zak transform of , but it turns out that the approach does not quite work when doing this for topological reasons (relating to the fact that scalar quasiperiodic functions of mean zero are forced to have zeroes), and so the authors used the slightly denser lattice instead , which relates to a vector-valued version of the Zak transform taking values in rather than . Applying this transform, the eigenvalue problem can be transformed by standard calculations to a "vector cocycle problem"

where is a non-zero smooth quasiperiodic vector-valued function, is an irrational shift , and is a certain explicit matrix-valued function depending on the choices of , , and .

How to solve this equation? The motivating scenario here is if the matrix function was replaced by a rank one function

for some smooth vector-valued function of unit magnitude. Then one could solve the equation by taking and . It is not possible to make the function exactly of this form, but through some numerical computation and clever AI-assisted guesswork, the authors were able to find a choice of and , and that made approximately equal to a rank one function of this form, in fact getting a uniform estimate

As it turns out, such an approximation is sufficient to run a...