Determine the global scaling of the CAID support cardinality

Determine the precise asymptotic scaling of the support cardinality of the capacity-achieving input distribution for the amplitude-constrained AWGN channel as the amplitude constraint A grows.

Background

For the amplitude-constrained AWGN channel, the capacity-achieving input distribution is known to be discrete and finitely supported, but its exact support structure is not characterized for general amplitude constraints. The paper notes that existing non-asymptotic bounds place the support cardinality between orders A√(log A) and A², leaving the precise scaling unresolved.

References

In general, both the exact value of the capacity $C(A)$ and the precise structure of the capacity-achieving input distribution (CAID) $P_{X\star}$ remain unknown.

— On the Evolution of the Capacity-Achieving Input Support for the Amplitude-Constrained AWGN Channel  (2609.10039 - Barletta et al., 9 Sep 2026) in Section 1, Introduction; Section 1.1, Past Work