Erdos 1954 results additive number theory

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question_p853: The paper's closing question asks whether some sequence b_j with fewer than c'_10 x/log x terms up to x has every sufficiently large integer of the form 2^l + b_j.

theorem_1: Erdős's construction of a sequence b_j with fewer than c_5 (log n)^2 terms up to n, for every n, such that every sufficiently large integer is a prime plus some b_j, improving the (log n)^3 that Lorentz's general bound gives.

theorem_2: Erdős's random construction of a sequence a_i of positive lower density such that every sequence b_j with all sufficiently large integers of the form a_i + b_j has more than c_7 (log n)^2 terms up to n, so the (log n)^2 bound for such sequences cannot be lowered in general.

theorem_p851: Erdős's proof of the conjecture, recorded in the paper from Volkmann, that two pseudorational sequences in the sense of Buck and Volkmann can have a sumset that is not pseudorational; the sumset built has upper density 1 and lower density 0.

theorem_p853: Erdős's closing remark that some sequence b_j with fewer than c_10 x^{1/2} terms up to x has every large integer of the form l^2 + b_j, with the analogous bound c_k x^{1-1/k} for k-th powers, answering a question of Lorentz.

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P. Erdős: Some results on additive number theory, Proc. Amer. Math. Soc. 5 (1954), 847--853 (MR 16,336b; Zentralblatt 56,270). DOI: https://doi.org/10.1090/S0002-9939-1954-0064798-9.

On a question of Lorentz, Theorem 1 (p. 847) constructs a sequence b_1 < b_2 < ... with N(b_j,n) < c_5 (log n)^2 for all n such that every sufficiently large integer is of the form p + b_j with p prime, improving the (log n)^3 that Lorentz's general bound (1) gives; the paper notes that N(b_j,n) must exceed c_3 log n, and leaves open whether Theorem 1 is best possible (p. 849). The proof (pp. 848--849) builds the sequence from blocks given by a Lemma (p. 848) proved by counting, with the Hoheisel--Ingham theorem on primes in short intervals. For sequences a_i of positive lower density, (1) gives b_j with N(b_j,n) < c_6 (log n)^2, and Theorem 2 (p. 848) shows this is best possible in general: there is a sequence a_i with N(a_i,n) > alpha n for all large n such that any b_j whose sums with it cover all large integers satisfies N(b_j,n) > c_7 (log n)^2. Its proof (pp. 849--851) takes a random set A_t, keeping each integer of the intervals (8^k, 2·8^k] with probability 1/2, and uses the Borel--Cantelli lemma. The paper also proves (pp. 851--852) a conjecture it attributes in footnote 5 to Volkmann (printed "Volkman"): the sum of two pseudorational sequences, in the sense of Buck and Volkmann (p. 848), need not be pseudorational. It closes with remarks stated without proof: a characterization, by branching systems of residues modulo k!, of the sequences S of density 0 for which some pseudorational B makes S + B not pseudorational (pp. 852--853), and, answering a question of Lorentz recorded on p. 849, a sequence b_j with N(b_j,x) < c_10 x^{1/2} such that every large integer is l^2 + b_j, with an analogue c_k x^{1-1/k} for k-th powers; it then asks whether some b_j with N(b_j,x) < c'_10 x/log x has every sufficiently large integer of the form 2^l + b_j (p. 853).

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Read status. Claims checked: Theorems 1 and 2, the Lemma, the pseudorational theorem and the remarks of p. 853 were read clause by clause on the printed pages. The proofs of Theorems 1 and 2 and of the pseudorational theorem were read but not checked step by step; the remarks of pp. 852--853 have no proofs in the paper beyond the construction named for squares.

Bears on. #32: Theorem 1 gives a set A with |A ∩ [1,N]| < c_5 (log N)^2 such that every large integer is p + a, the bound the problem's first question asks to improve to o((log N)^2); it answers none of the problem's three questions. Theorem 2 concerns a sequence of positive lower density, which the primes are not, and gives no bound for the problem. #33: the remark of p. 853 gives a set A with |A ∩ [1,N]| < c_10 N^{1/2} and every large integer of the form n^2 + a, so the problem's smallest limsup is finite; the paper names no value for c_10, and the remark determines neither of the problem's questions. #221: the paper's closing question (p. 853) is the question of the problem; the paper does not answer it.

Results. Theorem 1 (p. 847, with the Lemma of p. 848); Theorem 2 (p. 848); the pseudorational theorem (p. 851, unnumbered, with the definition of p. 848); the remark on squares and k-th powers (p. 853, unnumbered); the question on powers of 2 (p. 853, unnumbered). The residue bound (2) of p. 849 is noted on the Theorem 2 page.

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