The set of values of any finite iteration of Euler’s $\varphi $ function contains long arithmetic progressions

Assuming the validity of Dickson’s conjecture, we show that the set of values of iterated Euler’s totient $\varphi $ function $\varphi \circ \cdots \circ \varphi $ ($n$ times) contains arbitrarily long arithmetic progressions with an explicitly given common difference $D_a$ depending only on $a$. This extends a previous result (case $a = 1$) of Deshouillers, Eyyunni and Gun. In particular, this implies that this set has upper Banach density at least $1/D_a \gt 0$.