The sum-of-two-squares conjecture for the sets $A_{Q,r}$

Let $\mathbf t=(t_0,\ldots,t_{d-2})$ and

$$Q(\mathbf t):=\frac{N}{q^2}-\sum_{i=0}^{d-2}\left(t_i+\frac{b_i}{q}\right)^2,$$

where $N,b_0,\ldots,b_{d-2}$ are integers, $N\equiv\sum_{i=0}^{d-2}b_i^2\pmod q$, and $\gcd(N,q)=1$. For $r>0$, let

$$A_{Q,r}:=\{\mathbf t\in\mathbb Z^{d-1}:Q(\mathbf t)\in\mathbb Z,\ |\mathbf t|<r,\text{ and }Q(\mathbf t)\geq0\}.$$

The sum-of-two-squares conjecture. There exist constants $\gamma>0$ and $C_\gamma>0$, independent of $Q$ and $r$, such that if

$$|A_{Q,r}|>C_\gamma(\log N)^\gamma$$

for some $r>0$, then $Q$ expresses a sum of two squares inside $A_{Q,r}$. This arithmetic conjecture is used to establish polynomial-time termination of the paper's conditional lifting algorithm.