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Let $\theta$ be a positive irrational number and $S=\{\theta n^2+m^2: n, m\in \mathbb{N}\}$. The elements of $S$ can be written as a sequence of strictly increasing numbers $\{s_n\}$. My question is what is known about the difference $s_{n+1}-s_n$? Is there an estimate like $s_{n+1}-s_n\ge \frac{c}{n^\sigma}$ with som $\sigma\in (0,1)$, at least under some conditions on $\theta$?

Let $\theta$ be a positive irrational number and $S=\{\theta n^2+m^2: n, m\in \mathbb{N}\}$. The elements of $S$ can be written as a sequence of strictly increasing numbers $\{s_n\}$. My question is what is known about the difference $s_{n+1}-s_n$? Is there an estimate like $s_{n+1}-s_n\ge \frac{c}{n^\sigma}$ with some $\sigma\in (0,1)$, at least under some conditions on $\theta$?