Under dyadic regularity, log A(n) = sqrt((log 2) log n) + (1/4) log log n + O_dens(1), with dyadic nonconcentration also established.
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Proves density-one lower bound g_k(n) ≥ (3(k-1)/log 12 - ε) log n for almost all n and pointwise upper bound g_k(n) ≤ (k-1)log2 n + log2 log n + O_k(1).
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A Resolution of Erd\H{o}s Problem 731 under Dyadic Regularity
Under dyadic regularity, log A(n) = sqrt((log 2) log n) + (1/4) log log n + O_dens(1), with dyadic nonconcentration also established.
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Prime-Power Rarefaction and a Density-One Lower Bound for Erd\H{o}s Problem 400
Proves density-one lower bound g_k(n) ≥ (3(k-1)/log 12 - ε) log n for almost all n and pointwise upper bound g_k(n) ≤ (k-1)log2 n + log2 log n + O_k(1).