Cilleruelo 1995 b 2 g sequences whose terms

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theorem_1: Cilleruelo's theorem that the sequence in the Erdős-Rényi theorem can be taken among the squares: for every epsilon > 0 there are a natural number g and a B_2[g] sequence of squares a_k^2 with a_k << k^{1+epsilon}, g depending on epsilon.

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Javier Cilleruelo, B_2[g] sequences whose terms are squares. Acta Mathematica Hungarica 67 (1995), no. 1-2, 79-83. doi:10.1007/BF01874521.

The note transfers the Erdos-Renyi theorem on B_2[g] sequences into the sequence of perfect squares. Theorem 1 states that for every epsilon > 0 there is a natural number g and a B_2[g] sequence of squares {a_k^2} with a_k << k^{1+epsilon}, i.e. the squares themselves are O(k^{2+2epsilon}); the natural number g is allowed to depend on epsilon. The proof takes any g > 2/epsilon and builds {a_k} with a_k << k^{1+epsilon/2} in which every n >= n(epsilon) has at most g representations n = a_j^2 + a_k^2 with a_j <= a_k. The method is Erdos's probabilistic construction: a probability space on integer sequences with P(n in A) = n^{-c}, c = epsilon/(2+epsilon), so that almost surely a_j ~ (1-c) j^{1/(1-c)}, plus a Borel-Cantelli argument bounding the number of representations n = a_j^2 + a_k^2. The paper notes that the full sequence of squares is a B_2[g] sequence for no g, since the number of representations as a sum of two squares is unbounded. Pages cited here are those of the author-typeset version read (pp. 1-6): the setting on p. 1, Theorem 1 on p. 2, the proof on pp. 2-5.

Source: https://doi.org/10.1007/BF01874521. The copy read for this card is an author-typeset version, which prints no notice; the version of record's Springer article page shows only the site footer "© 2026 Springer Nature" and names no license (https://link.springer.com/article/10.1007/BF01874521, read 2026-10-02), and does not govern that version; the term is unstated.

Bears on. #158: Theorem 1 (p. 2) gives infinite B_2[g] sequences of squares, but g depends on epsilon (the proof needs g > 2/epsilon) and the number of terms up to N is only

N^{1/(2+2 epsilon)} (computed on the result page), below N^{1/2}. The problem fixes g = 2 and asks about liminf A(N)/N^{1/2}; the theorem neither answers nor refutes it.

Results. Page numbers are those of the version read (pp. 1-6).

• Theorem 1 (p. 2; proof pp. 2-5): for every epsilon > 0 there exist a natural number g and a B_2[g] sequence of squares {a_k^2} with a_k << k^{1+epsilon}. The proof takes any natural number g > 2/epsilon and gets a_k << k^{1+epsilon/2} with at most g representations of every n >= n(epsilon). The page also records the remark (p. 1) that the full sequence of squares is B_2[g] for no g.
• Probabilistic setup (p. 2, quoted from Halberstam and Roth, Sequences, pp. 142-144): the Erdos measure on integer sequences with P(n in A) = n^{-c}; with c = epsilon/(2+epsilon), almost surely a_j ~ (2/(2+epsilon)) j^{1+epsilon/2}. Not a result of the paper; described in the proof pointer of the Theorem 1 page.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.