Three-rational-incidence saving conjecture

Let $N$ be a positive integer. There is a constant $\beta>0$ such that, for all sufficiently small $\alpha>0$, the following holds. Assume

$$Q2^l\lesssim N,\qquad Q\geq1,\qquad 2^l\geq1,\qquad Q\leq N^{2/3}.$$

For each $t,t'$ satisfying $|t-t'|\gg1/(NQ2^l)$, consider triples $(a_1,a_2,a_3,q_1,q_2,q_3)$ with $q_i\sim Q$, $|a_i|<q_i$, ${\operatorname{gcd}}(a_i,q_i)=1$, and

$${\operatorname{gcd}}(q_1,q_2)\leq N^\alpha,\qquad {\operatorname{gcd}}(q_1,q_3)\leq N^\alpha,$$

and satisfying

$$\left|\frac{a_1}{q_1}-\frac{a_2}{q_2}-t\right|\lesssim\frac1{NQ2^l},\qquad \left|\frac{a_1}{q_1}-\frac{a_3}{q_3}-t'\right|\lesssim\frac1{NQ2^l}.$$

Three-rational-incidence saving conjecture. The number of such solutions is

$$\lessapprox N^\alpha+\frac{Q^3}{2^lN^{1+\beta}}.$$

This conjecture would provide a factor of $N^\beta$ saving over the corresponding one-equation solution count and is proposed as a route to improvements for the periodic Schrödinger maximal-function estimate.