Growth beyond the limiting exponent of the recursive method

Establish a growth bound with exponent strictly greater than $\phi=(1+\sqrt{5})/2$ for iterated additive expressions of images of finite sets under linearly independent analytic functions, beyond the exponent approached by the recursive function $\phi(n)$.

Background

Theorem \ref{growthforsomef} produces, for a sufficiently large family of analytic functions with linearly independent derivatives, a lower bound with exponent ϕ(n)\phi(n), where the recursive sequence converges to the golden ratio ϕ=(1+5)/2\phi=(1+\sqrt5)/2 from below. The author explicitly states that the argument does not establish growth beyond this limiting exponent, while it can approach the exponent arbitrarily closely.

References

Noting that the sequence $\phi(n)$ converges to $\phi:=\frac{1+\sqrt{5}}{2}$ from below, we are unable to prove growth beyond $|A|\phi$, but are able to get arbitrarily close by considering enough functions and long enough sums.

— Additive growth amongst images of linearly independent analytic functions  (2503.03690 - Mansfield, 5 Mar 2025) in Section 1, immediately after Theorem 1.6 (Theorem \ref{growthforsomef})