How many 3-term arithmetic progressions can there be if there are no longer ones

Journal Article · Sat Oct 01 00:00:00 EDT 1977 · Am. Math. Mon.; (United States)

DOI:https://doi.org/10.2307/2321017· OSTI ID:5114772

Simmons, G J; Abbott, H L

Let script-A/sub k/(n) be the set of n-term nonnegative integer sequences that contain no k-term arithmetic progression (AP) as a subsequence. Denote by f(A) the number of 3-term APs in the sequence A. Define f/sub k/(n) = max f(A) for A an element of script-A/sub k/(n) and s/sub k/ = lim/sub n..-->..infinity/ log f/sub k/(n)/log n. It is shown that f/sub 4/(n) greater than or equal to n/sup 1.623/ infinitely often and that s/sub k/ ..-->.. 2 as k ..-->.. infinity. The question of determining lim/sub n..-->..infinity/f/sub 4/(n) and, more generally, of determining s/sub k/ is addressed. (RWR)

• OSTI ID: 5114772
• Journal Information: Am. Math. Mon.; (United States), Vol. 84:8
• Country of Publication: United States
• Language: English

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990200* - Mathematics & Computers