The density of zeros of forms for which weak approximation fails

Abstract:

The weak approximation principle fails for the forms ${x^3} + {y^3} + {z^3} = k{w^3}$ , when $k = 2$ or 3. The question therefore arises as to what asymptotic density one should predict for the rational zeros of these forms. Evidence, both numerical and theoretical, is presented, which suggests that, for forms of the above type, the product of the local densities still gives the correct global density.