MathDB

Wu–Zhan Problem 2 on generalized Riemann-type graph dimensions

Let gg be a nonconstant real-valued trigonometric polynomial, and let S(a;q)S(a;q) denote the associated square-class chirp functional at the rational point a/qa/q, as defined in the Wu–Zhan formulation. Is it possible that S(a;q)=0S(a;q)=0 for every rational number a/qa/q? Equivalently, must there exist coprime integers a,qa,q with q≥1q\ge 1 such that S(a;q)≠0S(a;q)\ne 0?

References

Primary source

arXiv

Additional references

fractal-geometryharmonic-analysisnumber-theory
Posted

Progress summary

Refreshed
Claimed solved

An unrefereed 2026 preprint claims to settle the problem negatively, but the claim has not been independently checked.

Wu–Zhan Problem 2 concerns whether the relevant chirp functional can vanish at every rational for a nonconstant real trigonometric polynomial. A reported resolution says it cannot, giving the advertised graph-dimension formula and a negative answer.

Known results

  • A 2026 preprint proves conditional box-dimension bounds, including the value 74−δ2\frac{7}{4}-\frac{\delta}{2} under a nonvanishing weighted Fourier-coefficient condition, and explicitly leaves the all-vanishing case unresolved.
  • The same work shows that vanishing of the arithmetic sum alone does not determine the dimension, since translation symmetries can preserve the expected dimension.

August 2026 claimed resolution

On August 25, 2026, a report described A Rational-Level Criterion on Box Dimension of the Graph of Generalized Riemann-Type Functions as proving that the chirp functional cannot vanish at every rational for the specified class, thereby settling the problem negatively. This is an unrefereed claim, and no independent verification was found.

Current status (as of August 2026): The problem is claimed solved by the August 2026 preprint, but the resolution remains unverified; the earlier conditional work does not by itself settle the vanishing case.

Sources

Solutions 0

No solutions have been posted yet.