Number of integers represented by families of binary forms. II: Binomial forms (Q6595587)

For a given \(d \geq 3\), let \(\mathcal{E}_d\) be a finite subset of \((\mathbb{Z} \setminus \{0\}) \times (\mathbb{Z} \setminus \{0\})\) and let \(\mathcal{F}_d\) be the set of binary binomial forms \(F_{a,b,d}(X,Y) = aX^d + bY^d\) with \((a,b) \in \mathcal{E}_d\). Next, for \(d \geq 3\) and \(m \in \mathbb{Z}\), let us consider the sets:\N\[\N\mathcal{G}_{\geq d}(m) = \left\{ (d',a,b,x,y) \ \middle|\ m = ax^{d'} + by^{d'}, d' \geq d, (a,b) \in \mathcal{E}_{d'}, (x,y) \in \mathbb{Z}^2, \max\{|x|, |y|\} \geq 2 \right\},\N\]\Nand\N\[\N\mathcal{R}_{\geq d} = \left\{ m \in \mathbb{Z} \ \middle|\ \mathcal{G}_{\geq d}(m) \neq \emptyset \right\}.\N\]\NFinally, for a positive integer \(N\), let us write \(\mathcal{R}_{\geq d}(N) = R_{\geq d} \cap [-N, N]\). Moreover, we assume that any two form \(\mathcal{E}_{d}\) are not isomorphic (the condition which can be easily stated in the terms of quotients of the coefficients).\N\NIn this paper the authors prove that under mild conditions on the cardinality of \(\mathcal{E}_{d}\), the set \(\mathcal{G}_{\geq d}(m)\) is finite for \(d\geq 4\) and \(m\in\mathbb{Z}\setminus\{0, 1\}\). Moreover, there are positive constants \(C, \alpha\in\mathcal{R}\) with \(c<2/d\) such that \(\sharp\mathcal{R}_{\geq d}(N)=C\cdot N^{2/d}+O(N^\alpha)\).