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Statement

Printed p. 6201, after the construction: "As for an upper bound, it is easy to prove that the proportion cannot exceed $2/3$, moreover this holds if we exclude only $a_i=a_j+a_{j+1}$ (and for this case it is the best possible)". No proof is given.

The report that follows: "Later I learned from D. Coppersmith and Steven Phillips (Thomas J. Watson Research Center, Yorktown Heights, NY, USA) that they had rediscovered my result above and improved it; they have a construction giving $13n/24+O(1)$. They also improved the upper bound to $2/3-1/3584$." The figures $13n/24$ and $1/3584$ were read at 300 dpi. The site's commentary on Problem 867 and the catalog's Lean file print the Coppersmith--Phillips upper bound as $(\tfrac23-\tfrac1{512})N+\log N$. The SIAM paper itself (SIAM J. Discrete Math. 9 (1996), no. 2, 173--177) is filed as coppersmith_phillips_1996_question_erdos_subsequence_sums; its abstract (printed p. 173, PDF p. 1) reads "impossible for $\epsilon=1/512$" and its Theorem 3.7 (printed p. 177, PDF p. 5) prints $2n/3-\lfloor n/512\rfloor+3\log_4n-1/2$, both read on the text layer, where the string $3584$ does not occur; the published figure is $1/512$, and the theorem is paged on theorem_3_7. Freud's $1/3584$ is his report as printed, not the paper's figure.

Source. R. Freud, Adding numbers, James Cook Mathematical Notes 6 (1993), issue 60, 6199--6202; printed p. 6201 is the right half of PDF p. 11 of the issue scan, read on the page image (150 dpi; 300 dpi for the two figures).

Read depth. Claims checked: the two paragraphs were read clause by clause on the page image. The $2/3$ bound is asserted without proof in the note and is not proved or checked here; the site's commentary gives an argument for $(\tfrac23+o(1))N$, an observation it credits to Sarosh Adenwalla.

Proof pointer

None in the note. For the $2/3$ bound the site's commentary sketches: if $|A\cap[x,2x]|=t$ then the $t-1$ sums of consecutive pairs are distinct members of $(2x,4x]$ outside $A$, so $|A\cap[x,4x]|\le2x+1$, and summing over the scales $4^{-i}n$ gives $|A|\le\tfrac23n+O(\log n)$ (an argument the site credits to Sarosh Adenwalla; not checked here).

Dependencies

None.

Bears on

• Problem 867: the note's remark, without proof, that the proportion of the problem's maximal $A$ in $\{1,\ldots,N\}$ cannot exceed $2/3$, and its report that Coppersmith and Phillips have a construction giving $13n/24+O(1)$ and the upper bound $2/3-1/3584$, the latter at variance with the published $\tfrac1{512}$; both figures are recorded on the problem page with their provenance.