Alexander nd density multiplicative structure sets integers

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Alexander, Ralph, Density and multiplicative structure of sets of integers. Acta Arith. 12 (1967), no. 4, 321--332. doi:10.4064/aa-12-4-321-332.

The paper develops a general framework of asymptotic densities induced by sequences of measures (Section 1), applies it through a multiplicative decomposition of a sequence (Section 2), and then sharpens the Erdős-Davenport results on multiplicative sets D(M), the multiples of a set M, and on division chains (Section 3). Theorem 3.6 (Erdős-Davenport, p. 328) shows every multiplicative set has a logarithmic density, equal to its lower natural density. Of the three division-chain theorems 3.10-3.12 (pp. 328-329), the first two (3.10 and 3.11) sharpen the Erdős-Davenport division-chain result: if C does not have zero logarithmic density then C contains a division chain q_1, q_1 q_2, q_1 q_2 q_3, ... in which each successive quotient is composed only of primes larger than (q_1...q_i)^{K_i} for any prescribed sequence K_i (Theorem 3.10). The third, Theorem 3.12, needs a stronger hypothesis: it gives the analog with primes larger than (q_1...q_i) raised to the power h(q_1...q_i), for an arbitrary h in the class Omega, when the lower logarithmic density is positive. Theorem 3.13 (p. 329) shows that Theorem 3.10 is best possible: for any Psi(n) tending to infinity there is a sequence of positive upper logarithmic density with no division chain whose quotients use only primes above d_i^{Psi(d_i)}. Theorem 3.19 (p. 331) concerns the irregularity of prime factors of almost all integers: for any positive integer r and suitable f, all but a set of zero natural density of integers n have prime factors p_1 < ... < p_t with p_{i+1} > f(p_i) for at least r values of i, complementing a related theorem of Erdős; Theorem 3.20 shows this fails for faster-growing f. This bears on problem 858 through the quantitative structure of gaps between consecutive prime factors of typical integers and the existence of division chains inside sets of positive density.

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Bears on. #858

Results to transcribe.

• Theorem 3.10: If C does not have zero logarithmic density then, for any prescribed positive K_1, K_2, ..., C contains a division chain q_1, q_1 q_2, q_1 q_2 q_3, ... in which q_{i+1} is composed entirely of primes greater than (q_1 ... q_i)^{K_i}.
• Theorem 3.12: If C has positive lower logarithmic density and h is an arbitrary element of the class Omega, C contains a division chain whose successive quotients use only primes greater than (q_1 ... q_i) raised to the power h(q_1 ... q_i).
• Theorem 3.13: For any arithmetic function Psi tending to infinity there is a sequence of integers of positive upper logarithmic density containing no division chain d_1 < d_2 < ... with d_{i+1}/d_i composed of primes greater than d_i^{Psi(d_i)}, so Theorem 3.10 is best possible.
• Theorem 3.19: For any positive integer r and f(n) = g(n)^{h(n)}, with g(n) the greatest prime factor of n and h in the class Omega, all but a set of zero natural density of integers n have prime factors p_1 < ... < p_t with p_{i+1}

  f(p_i) for at least r values of i.

• Theorem 3.20: If f(n) = n^{h(n)} with sum over n of 1/(n h(n) log n) convergent, then l(A(f)) > 0, showing Theorem 3.19 fails for faster-growing f.
• Theorem 3.6 (Erdős-Davenport): Every multiplicative set C has a logarithmic density, and it equals the lower natural density of C.