Growth beyond the golden-ratio exponent

Establish an additive-growth bound for images of finite sets under linearly independent analytic functions whose exponent exceeds the limiting value φ=(1+√5)/2 arising from the recurrence φ(1)=1 and φ(n)=1+1/(1+1/φ(n−1)).

Background

The paper proves that, given a finite family of n analytic functions with linearly independent derivatives, at least one function has an iterated sum-difference image of size at least a power |A|^{φ(n)}, where the exponents φ(n) increase toward the golden ratio φ=(1+√5)/2.

The author explicitly states that the method cannot prove an exponent beyond this limiting value, although it approaches the value arbitrarily closely by increasing the number of functions and the length of the sums. Thus, obtaining any bound with exponent strictly larger than φ remains an unresolved limitation identified in the paper.