Lorentz 1954 problem additive number theory

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greedy_interval_cover: Reconstructs Lorentz's greedy translation cover and the double count behind equations (2) through (4).

theorem_1: Reconstructs Lorentz's proof of the counting-function bound and its density-zero consequence for every infinite set.

theorem_2: Records Lorentz's finite residue-covering theorem at statement-and-pointer scope only.

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G. G. Lorentz, On a problem of additive number theory, Proceedings of the American Mathematical Society 5 (1954), no. 5, 838--841. The manuscript was received March 2, 1954, and the issue is dated October 1954. The registered DOI is 10.1090/S0002-9939-1954-0063389-3. The copy read for this card is the published scan, whose four physical pages are printed pp.838--841. No notice is printed on the scanned pages, the Crossref record names no license, and the publisher's copyright policy page (www.ams.org/publications/authors/ctp, read 2026-10-02) states that authors transfer copyright to the Society and names Creative Commons licenses only for its open-access series, every other right reserved.

For a set $A$ of positive natural numbers, write $A(n)$ for the number of $a\in A$ with $a\leq n$. Lorentz calls $A$ and $B$ complementary when $A+B$ contains every sufficiently large natural number. Throughout the reconstructed pages, $\log$ denotes the natural logarithm.

[[additive_bases/lorentz_1954_problem_additive_number_theory/theorem_1|Theorem 1]] proves that every infinite $A$ has a complementary set $B$ satisfying

$$B(n)\leq C\sum_{k=1}^{n}\frac{\log A(k)}{A(k)},$$

where $C$ is absolute and a term with $A(k)=0$ is replaced by $1$. The reconstructed proof includes the source's unlabeled [[additive_bases/lorentz_1954_problem_additive_number_theory/greedy_interval_cover|greedy interval-cover estimate]], the dyadic assembly, the exact reindexing, and the Cesàro argument giving $B(n)=o(n)$.

The printed proof compresses two endpoint details. The local cover needs $A(n-m+1)>0$; the reconstruction starts the dyadic construction only after $A(k)\geq3$, which discards finitely many target intervals and leaves both the cofinite conclusion and the displayed global bound intact. Printed equation (5) also uses equality signs where equation (4) and the subsequent block comparison provide upper bounds. The reconstruction records the printed notation and uses the inequalities required by the argument.

Besides $B(n)=o(n)$, printed p.840 draws two further unlabeled consequences of (1), recorded here without pages of their own. If $\liminf_{n\to\infty}\log A(n)/\log n=\alpha>0$, there is a complement $B$ with $\limsup_{n\to\infty}\log B(n)/\log n\leq1-\alpha$. If $A(n)\geq\alpha n$, (1) gives $B(n)\leq C\log^2n$; the paper remarks that (1) is likely best possible when only the growth of $A(n)$ is taken into account, and reports (pp.840--841) that Erdős showed by a probabilistic argument that the $\log^2n$ bound cannot be improved.

[[additive_bases/lorentz_1954_problem_additive_number_theory/theorem_2|Theorem 2]] is retained at statement-and-pointer scope only. It is a finite cyclic covering consequence of the same local estimate and is not used for Problem 31.

All four source pages were visually inspected. Theorem 1 is stated on physical p.1 / printed p.838; its proof occupies physical pp.1--3 / printed pp.838--840. Physical p.4 / printed p.841 contains the end of the surrounding discussion and Theorem 2. Native text was used only for navigation.

Source: AMS article record.

Bears on. #31: Theorem 1 gives every infinite $A$ a complement $B$ with $A+B$ containing every sufficiently large natural number and $B(n)=o(n)$, which is the problem's statement once $A$ is restricted to its positive elements if $0$ counts as a natural number. #32: the paper says nothing about the primes; Theorem 1 applied with $A$ the primes, together with Chebyshev's lower bound for the prime-counting function (an input not in the paper), gives a complement $B$ with $B(N)\ll(\log N)^3$. The problem asks for $o((\log N)^2)$, so this does not answer it.

Results transcribed.

• Theorem 1, printed pp.838--840: complete author reconstruction. The reported independent review is qualified below.
• The greedy interval-cover estimate and equations (2)--(4), printed pp.838--840: complete author reconstruction.
• Theorem 2, printed p.841: statement and source pointer only.

Current verification. The complete Theorem 1 reconstruction and its Problem 31 transfer are retained as author-recorded proof coverage. An independent mathematical review dated 2026-09-06 is reported, but its report is not filed with this source. The reported review therefore does not supply independent-review credit in this corpus. No formal-verification claim is made.

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