Linear quantitative dependence conjecture for Ramsey–Turán density regularity

Let $\mathcal{L}: c_1x_1 + \ldots +c_k x_k = 0$ be a homogeneous linear equation with $k \ge 3$, where $c_1,\ldots,c_k \in \mathbb{Z} \setminus \{0\}$ satisfy

$$c_1+\ldots+c_k \neq 0,$$

but there exists a nonempty subset $S\subseteq [k]$ such that

$$\sum_{s\in S} c_s = 0.$$

Here $d(\mathcal{L},\varepsilon)$ denotes the density parameter defined for the Ramsey–Turán analogue of Roth’s theorem. Quantitative dependence conjecture. Under these hypotheses,

$$d(\mathcal{L},\varepsilon)=\Theta(\varepsilon).$$

The paper establishes a lower bound of the form $\varepsilon^{O(k^2M)}$, where $M$ is the maximum absolute value of the coefficients, and determines the optimal exponent for Schur’s equation. The conjecture proposes the corresponding linear dependence in the stated degenerate-equation setting in general.