Optimal oscillation interval under uniform frequency gaps

Determine whether every nonzero real-valued almost periodic function with mean value zero, represented by a uniformly convergent Fourier series whose positive frequencies have consecutive gaps at least gamma, takes both positive and negative values on every interval of length at least 2π/γ.

Background

Theorem 3 proves that a nonzero real-valued almost periodic function with mean value zero and uniformly separated positive frequencies takes both positive and negative values on every interval of length at least 3π/γ. The authors state that this bound is not optimal and explicitly identify the weaker threshold 2π/γ as unresolved. Establishing this sharper estimate would be relevant to the subsequent application to vibrating strings, where the limiting interval length is important.

References

This result improves Proposition 3 of {\rm , reducing the condition on $|J|$ by a factor $\frac{3}{4}$, but it is not optimal and we still do not know if the weaker condition $ |J| \ge \frac {2\pi}{\gamma} $ is sufficient.

— Uniform oscillation properties of almost periodic functions  (2610.06515 - Haraux, 5 Oct 2026) in Remark following Theorem 3 (Section 3, “Gaps and oscillation”)