On the local structure of the set of values of Euler's $φ$ function

Assuming the validity of Dickson's conjecture, we show that the set $\mathcal{V}$ of values of the Euler's totient function $φ$ contains arbitrarily large arithmetic progressions with common difference 4. This leads to the question of proving unconditionally that this set $\mathcal{V}$ has a positive upper Banach density.

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