Nash 1989 sequences

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lemma_1: Nash's form of Erdos's B_2 argument: if the numbers D_l of elements of a sequence C of positive integers in the blocks ((l-1)N, lN], l = 1, ..., N, satisfy sum of D_l^2 << N, then liminf C(n)(log n)^{1/2}/n^{1/2} < infinity.

main_theorem: Nash's result, displayed as (3) with no theorem number: every B_4-sequence A of positive integers satisfies liminf A(n)(log n)^{1/4}/n^{1/4} < infinity, the B_4 analog of the B_2 bound the paper attributes to Erdos.

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Nash, John C. M., On {$B_4$}-sequences. Canad. Math. Bull. 32 (4) (1989), 446--449. The print carries "© Canadian Mathematical Society 1988." on its first page, every other right reserved.

Erdos had shown (the paper cites Stohr's 1955 survey) that the counting function A(n) of a B_2-sequence satisfies liminf A(n)(log n)^{1/2}/n^{1/2} < infinity; this short paper proves the analogous statement liminf A(n)(log n)^{1/4}/n^{1/4} < infinity for B_4-sequences, that is, sets in which every integer has at most one representation as a sum of four non-decreasing elements. The method reduces the claim to a counting condition (5) for the sumset 2A: Lemma 1 shows that if the numbers D_l of elements of a sequence C in the blocks ((l-1)N, lN], l = 1, ..., N, satisfy sum of D_l^2 = O(N), then C obeys the B_2-type liminf bound, and the author verifies this for C = 2A even though 2A is not itself a B_2-sequence. The verification bounds the number of 4-tuples (a_1,...,a_4) of elements of A up to N^2 with a_1+a_2-a_3-a_4 in (0,N], splitting them into a class controlled directly by the B_4 property (contributing at most 4N) and a class controlled by the count |T| of pairs of elements of A up to N^2 with difference in [1,N]. The bound A(N^2) = O(N^{1/2}), which follows from A(N) = O(N^{1/4}), and the comparison binom(|T|,2) <= (the 4-tuple count) give |T| = O(N^{1/2}) and close the argument; Cauchy's inequality enters through Lemma 1. The paper bears on problem 41 through the even case of the B_h question behind it: for B_4-sequences it gives liminf A(n)/n^{1/4} = 0 with a logarithmic saving, while problem 41 asks the same question for B_3-sequences (distinct triple sums), which this argument does not treat.

Source: https://doi.org/10.4153/cmb-1989-064-2.

Bears on. #41: the problem asks the $B_3$ case (all triple sums distinct) of the question whether an infinite $B_h$-sequence has $\liminf A(N)/N^{1/h}=0$. The main theorem settles the $B_4$ case with a logarithmic factor to spare and does not treat $B_3$-sequences.

Results. Main theorem (display (3), p. 446), the liminf bound for $B_4$-sequences; Lemma 1 (p. 447), the block-count criterion from Erdos's $B_2$ argument that the proof applies to $2A$.

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