Freud 1993 adding numbers problem p

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construction_p6199: Freud's four-block construction answering Erdős's question whether such a set can have significantly more than n/2 members, with the parameters, the deletion count and the infinite version with upper density 19/36, as printed in the 1993 note.

upper_bound_p6201: Freud's unproved upper-bound remark for sets with no member a sum of consecutive members, and his report that Coppersmith and Phillips rediscovered and improved his construction and the upper bound, with the figure 1/3584 as printed.

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R. Freud, Adding numbers. James Cook Mathematical Notes 6 (1993), issue 60 (January 1993), 6199-6202 (the heading of p. 6199 and the contents of p. 6183 print the title "Adding numbers" and the author's name "Ròbert Freud"; the site's [Fr93] adds "- on a problem of P. Erdős", which the issue does not print).

The copy read for this card is an image-only scan of the whole issue: 22 PDF pages, the cover (printed p. 6181) and then spreads of two printed pages (printed pp. 6182--6183 are PDF p. 2, 6198--6199 PDF p. 10, 6200--6201 PDF p. 11, 6202--6203 PDF p. 12), read on the page images at 150 dpi with a 300 dpi crop for the Coppersmith--Phillips figures. Read status: claims checked for the question as Freud states it, Pomerance's examples, the construction (A)--(D) with its conditions (i)--(iv), the deletion list I--V and the counts 76y-7 and 144y-12 (pp. 6199--6201), the upper-bound remark and the Coppersmith--Phillips report (p. 6201), the infinite version (pp. 6201--6202) and Erdős's companion note (p. 6203); the two counts were recomputed here from the printed figures; the verification that the construction has no consecutive sum in the set is the note's and is not checked here, and the 2/3 upper bound is asserted without proof. Statements are on construction_p6199 and upper_bound_p6201. No notice is printed on the issue cover or the last spread of the scan, and the hosting site states no license (https://webhomes.maths.ed.ac.uk/jcook/, read 2026-10-02); the term is unstated.

Erdos asked how large a set 1 <= a_1 < ... < a_k <= n can be if no a_i is the sum of two or more consecutive a_j's, and whether k can be much bigger than n/2. Freud gives an explicit four-block construction (blocks A-D built from parameters x, y around 2x, 3x, 4x and the interval [4x+4y+2, 8x+8y+4]) which after deleting the forbidden consecutive sums leaves 76y-7 elements up to n = 144y-12, so k = 19n/36 + O(1) is attainable; repeating the construction with rapidly growing y gives an infinite sequence with limsup A(n)/n = 19/36. He also asserts, without proof, that the proportion cannot exceed 2/3, and notes 2/3 is best possible if only the relations a_i = a_j + a_{j+1} are forbidden. The note records that Coppersmith and Phillips independently rediscovered and improved the construction to 13n/24 + O(1) and the upper bound to 2/3 - 1/3584 (as printed on p. 6201; the site's commentary on problem 867 and the catalog's Lean file print 2/3 - 1/512 for that bound). The SIAM paper is filed as coppersmith_phillips_1996_question_erdos_subsequence_sums; its Theorem 3.7 prints the bound 2n/3 - floor(n/512) + 3 log_4 n - 1/2 on printed p. 177 (PDF p. 5), read there on the page image and paged on theorem_3_7; its abstract prints epsilon = 1/512 on printed p. 173 (PDF p. 1, text layer), and the text layer of its five pages, nowhere prints 3584, so the published figure is the site's 1/512 and the 1/3584 is this note's alone. The scan is of the whole JCMN issue; page 6203 carries Erdos's companion note 'Adding numbers 2' asking whether a set in [1,n] all of whose consecutive-block sums are distinct must have k = o(n), and how large k can be. Freud's construction and upper bound are the quantitative content behind problem 867, his infinite version bears on problem 839, and the Erdos note on distinct consecutive sums is the Erdos-Harzheim question of problem 357, which Freud's note does not address.

Source: https://webhomes.maths.ed.ac.uk/cook/iss_no60.PDF.

Bears on. #839: the question as the note states it, printed p. 6199 (PDF p. 10), and the infinite version, printed pp. 6201--6202 (PDF pp. 11--12), page images: "Consider an infinite sequence. Let $A(n)$ denote the number of elements $\le n$. We can achieve $\limsup A(n)/n=19/36$ using the previous construction"; the note forbids sums of two or more consecutive members (the problem's consecutive $a_j$ with $j<i$), and a sequence with $\limsup A(n)/n=19/36$ has $\liminf a_n/n\le\tfrac{36}{19}$, which decides neither the problem's $\limsup a_n/n=\infty$ nor its logarithmic-density question; the site's key [Fr93]. #867: the construction (pp. 6199--6201) gives, for $N=144y-12$, sets of $\tfrac{19}{36}N-\tfrac23$ members, against the problem's $|A|\le N/2+O(1)$, the verification that no member is a consecutive sum being the note's; the unproved upper-bound remark ($2/3$) and the Coppersmith--Phillips report (p. 6201).

Results to transcribe.

• Construction (p. 6199-6201): For n = 144y-12 there is a set of 76y-7 integers in [1,n] with no element a sum of two or more consecutive elements, giving k = 19n/36 + O(1).
• Upper bound (p. 6201): The density of such a set is at most 2/3 (asserted, no proof); 2/3 is attained if only sums of two adjacent elements are forbidden.
• Infinite version (pp. 6201-6202): There is an infinite such sequence with limsup A(n)/n = 19/36, by repeating the construction with rapidly growing y.
• Remark (Coppersmith-Phillips, p. 6201, read at 300 dpi): Coppersmith and Phillips improved the construction to 13n/24 + O(1) and the upper bound to 2/3 - 1/3584 (the site prints 2/3 - 1/512).
• Erdos, 'Adding numbers 2' (p. 6203): Erdos and Harzheim ask whether a_1<...<a_k<=n with all consecutive-block sums distinct forces k = o(n), and whether k as large as n/(log n)^c is possible.

Overview

The note opens (p. 6199) with the question of how many members an increasing set in $[1,n]$ can have when no member is a sum of two or more consecutive members, and recalls Pomerance's 12-element example in $[1,20]$ and a family of size $n/2+2$. In the construction (pp. 6199–6201), conditions (i)–(iv) restrict which consecutive-block sums can occur, and the remaining conflicts lie in (D). The deletion classes I–V account for $8y+2$ values, leaving $4x+8y+1$ members; condition (iii), rather than (ii), gives the restriction $x\ge17y-2$, and equality there gives the $76y-7$ members in $[1,144y-12]$, hence the asymptotic proportion $19/36$.

For the infinite version (pp. 6201–6202), Freud repeats the finite construction at widely separated scales, choosing the next $y=T^2$ when $T$ is the sum of all preceding members. He specifies four intervals to delete at each new stage and asserts that this loses only about $4T$ members while preserving avoidance and $\limsup A(n)/n=19/36$, where $A(n)$ counts members at most $n$. The note has no numbered theorems or lemmas.

Relation to E839

This source bears on Problem 839.

In E839 notation, Freud's finite sets are initial segments $a_1<\cdots<a_k$ satisfying the same consecutive-block avoidance condition. Their $k/(144y-12)\to19/36$ rules out a universal density ceiling of $1/2$. The repeated construction gives an avoiding infinite sequence with positive upper counting density, so $\liminf a_n/n\le36/19$. It does not give bounded $a_n/n$ at every index, which is what a counterexample to E839's first question would require.

The construction thus gives a dense finite-block example and a positive-upper-density infinite example; it resolves neither universal assertion of E839.

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