Consider the diophantine equation $ $ x_1y_1^3 + \dots + x_s y_s^3 = 0. $ $
For fixed $ \mathbf{y}$ with coprime coordinates this is a $ s-1$ dimensional lattice $ \Lambda(\mathbf{y})$ . Let $ N(X)$ denote the number of integer points in this lattice with $ \max |x_i| \leq X$ . Then we should have $ $ N(X) – \frac{c(\mathbf{y})X^{s-1}}{d(\Lambda(\mathbf{y}))}\ll 1+ \frac{X^{s-2}}{\lambda_1 \dotsc\lambda_{s-2}},$ $
where the $ \lambda_i$ denote the successive minima. I would now like to sum this error term over coprime vectors $ \mathbf{y}$ with $ \max |y_i| \leq Y$ . Assume $ X \geq Y$ .
It may happen that $ \lambda_{s-1}$ is big (and conversely by Minkowski’s theorem the product of the other successive minima small) – that is of size $ d(\Lambda(\mathbf{y}))$ . I would like to show that this can not happen too often in the sense that the terms where $ \lambda_1$ is small do not occur on average often. Any input on how that might be achieved is greatly appreciated. Thank you!
