Decidability of zero tests for Diophantine approximation constants

Determine algorithmically whether the Diophantine approximation constants infₙ[nγ + α] and lim infₙ[nγ + α] are equal to zero, for the parameters α and irrational rotation angle γ considered in the Sturmian noisy-trace construction.

Background

The Sturmian case shows that allowing vanishing noise can produce traces that are not almost-periodic, unlike the traces generated by the paper’s noise-robust systems. The authors explain that noisy-trace queries can encode information about Diophantine approximation constants associated with an irrational rotation.

The stated unresolved issue is whether these constants can be algorithmically tested for equality to zero. This is presented as an open number-theoretic problem and motivates the connection between noisy-trace verification questions and Diophantine approximation.

References

The same arguments as above would allow us to encode the computations of Diophantine approximation constants (e.g., inf n[nγ + α], lim inf n[nγ + α], where even algorithmically determining whether these quantities are 0 remains an open number-theoretic problem) using queries such as, “given α and noise ν lower-bounded by 1/n, does tracesσ (α, ν) contain a trace x in which the factor 11 occurs (respectively, occurs infinitely often)?”

— Model Checking for Noisy Toric Words  (2610.23463 - Vahanwala, 16 Oct 2026) in Section 5, page XX:12