Weak ∈-nets and interval chains

We construct weak ∈-nets of almost linear size for certain types of point sets. Specifically, for planar point sets in convex position we construct weak 1/r-nets of size 0(r α(r)), where α(r) denotes the inverse Ackermann function. For point sets along the moment curve in R ^d we construct weak 1/r-nets of size r. 2 ^poly(α(r)), where the degree of the polynomial in the exponent depends (quadratically) on d. Our constructions result from a reduction to a new problem, which we call stabbing interval chains with j-tuples. Given the range of integers N = [l,n], an interval chain of length k is a sequence of k consecutive, disjoint, nonempty intervals contained in N. A jtuple p̄ = (p _i,..., p _j) is said to stab an interval chain C = I ₁...I _k if each p _i falls on a different interval of C. The problem is to construct a small-size family double-struck Z of j-tuples that stabs all k-interval chains in N. Let z ^(j) _k (n) denote the minimum size of such a family double-struck Z. We derive almost-tight upper and lower bounds for z ^(j) _k (n) for every fixed j; our bounds involve functions α _m(n) of the inverse Ackermann hierarchy. Specifically, we show that for j - 3 we have z ⁽³⁾ _k (n) = Θ(nα[k/2] (n)) for all k ≥ 6. For each j ≥ 4 we construct a pair of functions P ¹ _j(m), Q ¹ _j(m), almost equal asymptotically, such that z ^(j) _P′j(m)(n) = 0(nα _m(n)) and z ^(j) _Q'j(m)(n) = Ω(nα _m(n)).