On the distribution of products of members of a sequence with positive density (Q1409718)

Assume that \( \alpha > 0 \) and assume furthermore that \(a(i)\) (\(i \geq 1\)) is a set \(\mathcal A\) of natural numbers for which \(\liminf\{\text{card}\mathcal A \bigcap \{0,1,\ldots, M\}\}/M>\alpha\) as \(M\) increases. Let \(\mathcal B\) be the various distinct products \( b(n) \) arranged in ascending order of \( r \) of the \( a(i) \). The author conjectures that for unboundedly many \( b(n) \), \( b(n+1) - b(n) < c(\alpha,r) \), where \( c \) depends only upon \( \alpha \) and \( r \). He proves that this is true when \( r = 2 \), \( c(\alpha,r) \) then being \( k / \alpha^{4} \) for some constant \( k \). The conjecture is part of a complex of unsolved problems; in particular \textit{A. Sárközy} [Period. Math. Hung. 42, 17-35 (2001; Zbl 1062.11002)] conjectured that when \( r = 2 \), \( c(\alpha,r) \) may be taken to be \(1 /\alpha^{2} \). The theory has an extension to the integers mod \(p\). The author gives a complementary result concerning max \(|b(n+1)-b(n) |\) in this case.