The maximum VC-dimension conjecture for quadratic residues

Let $q$ tend to infinity through prime powers, and let $obreak\mathcal{S}_q$ denote the set of squares in the finite field $obreak\mathbb{F}_q$, viewed as a subset of its additive group. Write

$$\alpha_q=\frac{\operatorname{VCdim}_{\mathbb{F}_q}(\mathcal{S}_q)}{\log_2 q},\qquad \underline{\alpha}=\liminf_{q\to\infty}\alpha_q.$$

Maximum VC-dimension conjecture. One has

$$\underline{\alpha}=1.$$

Equivalently, $obreak\operatorname{VCdim}_{\mathbb{F}_q}(\mathcal{S}_q)=(1+o(1))\log_2q$ as $q\to\infty$; in particular, the squares have asymptotically the maximum possible VC-dimension. The paper presents this as a natural expectation motivated by the multiplicative structure of the squares and the pseudorandomness of their interaction with addition; the conjecture is not resolved in the supplied text.