Kloosterman $L^1$ anti-concentration conjecture

Let $q$ be an odd prime power, let $\chi$ be the character used to define the classical Kloosterman sum

$$K(a)=\sum_{r\in\mathbb F_q^\times}\chi\left(r+\frac{a}{r}\right),\qquad a\in\mathbb F_q,$$

and, for a nonempty set $T\subseteq\mathbb F_q^\times$, let

$$\mathcal P(T)=\left\{\lambda=(\lambda_t)_{t\in T}:\lambda_t\geq0,\ \sum_{t\in T}\lambda_t=1\right\}.$$

Kloosterman $L^1$ anti-concentration conjecture. For every $0<\alpha<\frac12$, there exists a constant $c_\alpha>0$ such that, for every odd prime power $q$, if $\varnothing\ne T\subseteq\mathbb F_q^\times$ satisfies

$$|T|\leq\alpha(q-1),$$

then every probability measure $\lambda\in\mathcal P(T)$ satisfies

$$\Phi_T(\lambda):=\frac1{q-1}\sum_{s\in\mathbb F_q^\times}\left|\sum_{t\in T}\lambda_tK(st)\right|\geq c_\alpha.$$

The conjecture is the analytic input for the paper's semidefinite approach to the two-set Erdős–Falconer distance problem. The paper reports constant-scale evidence for some measures and structured supports, while proving unconditional bounds at the $q^{-1/2}$ scale in general and at the $q^{-1/3}$ scale when $0<\alpha<\frac12$; the constant-scale assertion remains open.