The Density of Zeros of Forms for which Weak Approximation Fails

The weak approximation principal fails for the forms x + y + z = kw, 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. Let f(x1, . . . , xn) ∈ Q[x1, . . . , xn] be a rational form. We say that f satisfies the weak approximation principle if the following condition holds. (WA): Given an ε > 0 and a finite set S of places of Q, and zeros (x1 , . . . , x ν n) ∈ Qν of the form f, we can find a rational zero (x1, . . . , xn) of f such that, |xi − xi |ν < ε, for 1 ≤ i ≤ n and ν ∈ S. Alternatively, we may write X(K) for the points on the hypersurface f = 0 whose coordinates lie in the field K, and consider the product