An elliptic analogue of Roth's theorem (Q2565521)

Let \(E\) be an elliptic curve defined over the number field \(K\) such that the \(\mathbb Z\)-rank of the group \(E(K)\) is \(r\). The author defines a height \(h\) on \(E(K)\) and for every place \(v\) of \(K\) a distance \(\text{dist}_v\). Let \(S\) be a finite set of places of \(K\) and let \((\lambda_v)_{v\in S}\) be a family of positive real numbers satisfying \(\sum_{v\in S}\frac{[K_v:{\mathbb Q}_v]}{[K:{\mathbb Q}]}\lambda_v=1\). Theorem 1. For every \(0<\varepsilon<2^{-14}\) put \[ A_1=1+\frac{188}{\log| \log \varepsilon| }, \quad B_1=4\varepsilon^{-r}, \] \[ A_2=\frac{r}{2}(\log r+\log\log| \varepsilon| +17),B_2=r^2(\log r+\log| \log \varepsilon| +83). \] Then for \(i\in\{1,2\}\) the set of points \(x\in E(K)\) satisfying the system of inequalities: \[ \text{dist}_v(x,0)<\exp\{-\lambda_v(\varepsilon h(x)+(\eta+5)\varepsilon^{-A_i})-2m_v-16\}\quad (v\in S) \] is finite with cardinal bounded by \(2B_i\varepsilon^{-\frac{1}{2}} | \log\varepsilon| ^2(499\varepsilon^{-\frac{1}{2}})^r.\) Theorem 2. For every \(0<\varepsilon<2^{-14}\) the set of points \(x\in E(K)\) satisfying the system of inequalities: \[ \text{dist}_v(x,0)<\exp\{-\lambda_v\varepsilon h(x)-2m_v-16\}\quad (v\in S) \] is finite with cardinal bounded by: \[ (\text{card}(E(K)_{\text{tor}})+ \varepsilon^{\frac{-1}{2}}) \Biggl(1+\frac{15(\eta+4)^{\frac{1}{2}}} {\widehat{h}_{\min}^\frac{1}{2}}\Biggr)^r (\varepsilon^{-\frac{1}{2}- \frac{92}{\log| \log\varepsilon| }})^r, \] where \(\widehat{h}_{\min}\) is the least non-zero value of the Neron-Tate height on \(E(K)\).