Bourgain 1997 estimates related sumfree subsets sets integers

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display_8_4: Bourgain's § 8 fact that no constant fraction of a finite set of positive integers can always be kept k-sum-free for every k: if s_k(A) > delta_k |A| holds for all finite A, then delta_k tends to 0, shown by sets built from blocks of multiples of factorials and an unproved circle-method lemma.

proposition_1_3: Bourgain's bound S(B) ≥ (|B| + 2)/3 for the largest sum-free subset of a set B of positive integers, proved for |B| ≥ 3 by a case analysis on the three smallest elements with Erdős's rotation argument written as a Fourier minorization; the (n + 2)/3 in the chain of lower bounds for Problem 792.

proposition_1_4: Bourgain's lower bound S(B) >= |B|/3 + c_1 (log |B|)^{-1} times the L^1 norm of the sum of cos 2 pi k theta over k in B, for the largest sum-free subset of a finite set B of positive integers; the L^1 route to Problem 792 that Bedert's 2025 bound develops.

proposition_1_7: Bourgain's bound S_3(B) > |B|/4 + c log |B| / log log |B| for the largest subset A of a finite set B of positive integers with (A + A + A) ∩ A empty, proved with two asymmetric arcs about 1/4 and -1/4 and the paper's version of the McGehee-Pigno-Smith L^1 estimate.

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J. Bourgain, Estimates related to sumfree subsets of sets of integers, Israel Journal of Mathematics 97 (1997), 71--92, DOI 10.1007/BF02774027 (the running head prints "ISRAEL JOURNAL OF MATHEMATICS 97 (1997), 71--92"; the DOI is the publisher's, from the Crossref record read, and is not printed on the scan); the author at the Institute for Advanced Study; received March 7, 1995 (p. 71). Cited as [Bo97] on the problem page. Its four references (p. 92) are Alon and Kleitman, Sum-free subsets (1990), filed as alon_1990_sum_free_subsets; the author's $\ell^1$-sequences generated by Sidon sets (Proc. London Math. Soc. 29 (1984), 283--288); Erdős, Extremal problems in number theory (1965), filed as erdos_1965_extremal_problems_number_theory; and McGehee, Pigno and Smith, Hardy's inequality and the $L^1$-norm of exponential sums (Ann. of Math. 113 (1981), 613--618). The edition cited is the publisher's version of record at https://doi.org/10.1007/BF02774027; no preprint or repository version is known here.

The copy read for this card is the publisher's scan of the printed article: 22 pages, printed pp. 71--92 = PDF pp. 1--22 (printed p. $n$ is PDF p. $n-70$), a 2007 scan (the copy's metadata names a TIFF source and a November 2007 creation date) with an OCR text layer that reads the prose and locates the labeled statements but garbles the displays (fractions, subscripts, inequality signs, the sums and norms come out as scattered characters). Provenance: obtained from the publisher on 2026-09-22 as a DRM-free per-article PDF through the library's acquisition, from https://doi.org/10.1007/BF02774027 (resolving to the article at link.springer.com); 501,805 bytes. No notice is printed on the scan; the publisher's article page (https://link.springer.com/article/10.1007/BF02774027, read 2026-10-02) shows "© Hebrew University" under Reprints and permissions, is paywalled, and names no Creative Commons license, every other right reserved.

Read status: claims checked for the abstract, displays (1.1) and (1.2), Propositions 1.3, 1.4 and 1.7 with (1.5) and (1.6) (printed pp. 71--72), the harmonic-analysis formulation of § 2 (p. 73), and the proof of Proposition 1.3 in § 3 (pp. 74--76), each read clause by clause on the page images of PDF pp. 1--6 on 2026-09-22; the opening of § 8 with Lemma 8.5 (p. 89, PDF p. 19), the construction on p. 90 (PDF p. 20) and the references (p. 92, PDF p. 22) were read on the page images. The proof of Proposition 1.3 was read in full on the page images and its structure followed (the test-function minorization and the case split), but its numerical case bounds (3.9)--(3.23) were not recomputed. §§ 4--7 (pp. 77--88) and p. 91 of § 8 were read in the text layer for structure only, except Remark 1 of § 7 (p. 88), read on the page image, and no proof there was checked. On 2026-10-08 the statements of Propositions 1.4 and 1.7, (1.5), (1.6), Lemma 6.27 (p. 86) and § 8's (8.1)--(8.4) with Lemma 8.5 were read again clause by clause on the page images, and § 4 (p. 77), § 7 (pp. 87--88) and § 8 (pp. 89--91) were read on the page images for structure; their inequalities were not checked. Nothing here is independently reviewed.

Contents

• Abstract and § 1, Introduction (pp. 71--72, page images). The abstract opens with the definition, quoted: "A subset $A$ of the positive integers $\mathbb Z_+$ is called sumfree provided $(A+A)\cap A=\emptyset$." It then announces the paper's results: every finite $B\subset\mathbb Z_+$ has a sumfree subset $A$ with $|A|\ge\frac13(|B|+2)$, a slight improvement on Erdős [Erd] and Alon and Kleitman [A-K], proved by harmonic analysis refining Erdős's original approach; with $s_k(B)$ the maximum size of a $k$-sumfree subset $A$ of $B$, one whose $k$-fold sum $(A)_k=A+\cdots+A$ is disjoint from $A$, the same techniques give for instance $s_3(B)>\frac{|B|}4+c\frac{\log|B|}{\log\log|B|}$, improving the $s_k(B)>\frac{|B|}4$ that Erdős's argument yields; and any inequality $s_k(B)>\delta_k|B|$ valid for all finite $B\subset\mathbb Z_+$ forces $\delta_k\to0$ as $k\to\infty$, which the author says had been unclear. The abstract closes by calling the methods and the harmonic analysis questions they raise the most interesting part of the paper. The introduction recalls (1.1), Erdős's observation in [Erd] that any finite $B\subset\mathbb Z$ has a sumfree subset of size at least $\frac13|B|$, and (1.2), Alon and Kleitman's remark in [A-K] that the same argument gives strict inequality, hence a sumfree subset of size at least $\frac13(|B|+1)$; the paper's stated aim is to treat this and similar problems by harmonic analysis, bounding discrepancies with trigonometric-sum estimates, and its first illustration is a slight improvement of (1.2). Proposition 1.3 (p. 72, quoted): "$S(B)\ge\frac13(|B|+2)$, for any $B\subset\mathbb Z_+$. $S(B)$ denotes the maximum size of a sumfree subset of $B$." Proposition 1.4 (p. 72, quoted): "$S(B)\ge\frac{|B|}3+c_1(\log|B|)^{-1}\bigl|\sum_{k\in B}\cos2\pi k\theta\bigr|1$. Here $c_1$ is some fixed constant", with (1.5), from the solution of Littlewood's conjecture, $\bigl|\sum{k\in B}e^{2\pi ik\theta}\bigr|1\equiv\int_0^1\bigl|\sum{k\in B}\cos2\pi k\theta\bigr|,d\theta>c_2\log|B|$, cited to [M-P-S] for its proof; the paper remarks that in many cases Proposition 1.4 gives a larger gain over $|B|/3$. $S_3(B)$ is the size of the largest $A\subset B$ with (1.6) $(A+A+A)\cap A=\emptyset$, and Proposition 1.7 (p. 72, quoted): "$S_3(B)>\frac{|B|}4+c\frac{\log|B|}{\log\log|B|}$", presented as an improvement on the inequality $S_3(B)>|B|/4$, which the paper calls obvious, and as the technically most interesting part of the paper.
• § 2, Harmonic analysis formulation (p. 73, page image). $f$ is the indicator of the arc $J=\,]\tfrac12-\tfrac16,\tfrac12+\tfrac16[$ on $\mathbb T=\mathbb R/\mathbb Z$; a set $A\subset\mathbb Z$ all of whose dilates $nx$, $n\in A$, fall in $J$ for one $x\in\mathbb T$ is sumfree, since $J+J$ misses $J$, hence (2.1) $S(B)\ge\max_x\sum_{m\in B}f(mx)=\frac{|B|}3+\max_x\sum_{m\in B}(f-\tfrac13)(mx)$, Erdős's rotation argument. The Fourier expansion (2.2) writes $f-\frac13$ as $-\frac{\sqrt3}\pi\sum_{n\ge1}\frac{\chi(n)}n$ times the cosine at frequency $n$, where (2.3) $\chi(n)$ is $0,1,-1$ for $n\equiv0,1,2\pmod3$; a Möbius sieve over the primes up to $P$ ((2.4)--(2.6)) rewrites the combination of the sums $\sum_{m\in B}(f-\frac13)(mkx)$ over $k\mid P!$ with the signed weights $\mu(k)\chi(k)/k$ as $-\frac{\sqrt3}\pi\sum_{m\in B}\cos mx$ plus a sum over $n>1$ coprime to the primes up to $P$.
• § 3, Proof of Proposition 1.3 (pp. 74--76, page images). The quantity minorized is (3.1), $\max_x\sum_{m\in B}(f-\frac13)(mx) =\frac{\sqrt3}\pi\max_xF(x)$ with $F(x)=-\sum_{n\ge1,m\in B}\frac{\chi(n)}n\cos nmx$ (3.2). The elements of $B$ are $0<m_1<m_2<m_3<\cdots<m_N$, and $\gcd(B)=1$ is assumed without loss of generality. Case (I), $m_1>1$: with $j$ the least index such that $m_j\notin m_1\mathbb Z$ and the test function $G(x)=(1-\cos m_1x)(1-\cos m_jx)$ ($G\ge0$, $\int G=1$), (3.4)--(3.7) give (3.8) $\max F\ge\frac12+\frac12-\frac14=\frac34$ and (3.9) $(3.1)\ge \frac{\sqrt3}\pi\cdot\frac34=0{,}41\ldots>\frac13$. Case (II), $m_1=1$: if $m_2>2$, $G=1-\frac43\cos x+\frac13\cos2x$ gives (3.12) the same $\frac{\sqrt3}\pi\cdot\frac34>\frac13$; if $m_2=2$, $G=(1-\cos x)(1-\cos m_3x)$ and six subcases on $m_3$ ($m_3=3,4,5$; $m_3\ge6$ in each residue class modulo 3) give (3.15)--(3.23), each a numerical bound between $0{,}35\ldots$ and $0{,}39\ldots$, all $>\frac13$. Conclusion (p. 76): the case bounds (3.9), (3.12), (3.15), (3.16), (3.17), (3.19), (3.21) and (3.23) together give, in the paper's words, "for any $B\subset\mathbb Z_+$, $|B|\ge3$", the inequality (3.24) $\max_{x\in\mathbb T}\bigl[\sum_{m\in B}(f-\frac13)(mx)\bigr]>\frac13$, so by (2.1) $S(B)>\frac{|B|}3+\frac13$, and since $3S(B)$ is an integer, $S(B)\ge\frac{|B|}3+\frac23$, which is Proposition 1.3. The result page records a filing observation on the hypothesis $|B|\ge3$, which the statement on p. 72 omits.
• § 4, Proof of Proposition 1.4 (p. 77, text layer). Since $\int F=0$, (4.1) $\max F\ge\frac12\|F\|_{L^1}$; in the sieved form (2.6) the second term is bounded (4.3) by $C|B|P^{-1/2}$, and $P=|B|^2$ gives (4.4) $\|F\|_1>c(\log P)^{-1}\bigl\|\sum_{m\in B}\cos mx\bigr\|_1$.
• § 5, Further estimates on (2.6) (pp. 77--82, text layer). Lemma 5.1: for a finite $S\subset\mathbb Z_+$ and $P>(\log|S|)^2$, for every $K>1$ the sum of $|S\cap kS|$ over the $k\le K$ coprime to the primes up to $P$, divided by $K$ (not by the number of such $k$), is less than $CP^{-1/2}|S|\log|S|$, by a partition of these $k$ according to the quotient $k/q(k)$, $q(k)$ the largest prime divisor of $k$; a remark that the count of such $k$ alone "does not imply (5.2)"; Lemma 5.25 (an $\ell^2$ version for $f\in L^2(\mathbb T)$, $\|f\|_2\le1$, $P>(\log N)^4$) and Lemma 5.35 (a dyadic consequence for the sieved sum over $B$).
• § 6, The Littlewood conjecture revisited (pp. 83--86, text layer). The section reproduces the [M-P-S] proof of the Littlewood conjecture (6.1), $\|\sum_{n=1}^Ne^{im_nx}\|_{L^1(\mathbb T)}>c\log N$ for $m_1<\cdots<m_N$, with the adjustments the later sections need, ending in Lemma 6.22 and Lemma 6.27 ($P>(\log|B|)^{100}$, coefficients $|a_n|,|b_n|\le1$: an $L^1$ lower bound $c\log|B|$ for the sieved exponential sum).
• § 7, Proof of Proposition 1.7 (pp. 87--88, text layer). The approach is the one used to minorize $s_2(B)$, with the indicators $f_+$ and $f_-$ of two short arcs about $\frac14$ and $-\frac14$ (7.1) in place of $f$, and the $L^1$ estimates of § 6. Two remarks (p. 88). Remark 1 explains why the same device does not improve the lower bound for $s_2(B)$: a set $\Omega\subset\mathbb T$ with $|\Omega|=\frac13$ and $(\Omega+\Omega)\cap\Omega=\emptyset$, taken in place of the arc $]\frac12-\frac16,\frac12+\frac16[$, satisfies $\Omega=\mathbb T\setminus(\Omega-\Omega)$ by Kneser's theorem and is therefore symmetric. Remark 2 notes that the proof of Proposition 1.7 resembles [B].
• § 8, Further remarks (pp. 89--91; p. 89 and p. 90 on the page images, the rest in the text layer). $A\subset\mathbb Z_+$ is $k$-sum free if (8.1) $(A)_k\cap A=\emptyset$, $s_k(A)$ the largest size of a $k$-sum-free subset; (8.2) $s_k(A)>\frac1{k+1}|A|$ and an infinite version (8.3) both follow from the analogue of (2.1) for the arc $J=\bigl[\frac1{2(k-1)}-\frac1{2(k+1)},\frac1{2(k-1)}+\frac1{2(k+1)}\bigr]$, averaged over $x\in\mathbb T$. The section's main point is that an estimate (8.4) $s_k(A)>\delta_k|A|$ holding for every finite $A\subset\mathbb Z_+$ forces $\delta_k\to0$ as $k\to\infty$, shown by explicit examples. Lemma 8.5 ("an exercise on the circle method (we omit the proof)"): a set $A\subset\mathbb Z\cap[0,M]$ with $|A|>\delta M$ has integers $r,q<C(\delta)$ and an interval $I\subset[M,rM]$ of length $M$ with $q(I\cap\mathbb Z)\subset(A)_r$. The construction (pp. 90--91): $A_j=\{j!\,n\mid\frac N2\le n\le N\}$, $A=\bigcup_{j\le J}A_j$, and for $B\subset A$ with $|B|>\delta|A|$ three indices $j_0<j_1<j_2$ with $B$ dense in $A_{j_0}$ and $A_{j_1}$ and meeting $A_{j_2}$, so that for $k$ large depending on $\delta$ the $k$-fold sum $(B)_k$ covers $A_{j_2}$ and $(B)_k\cap B\ne\emptyset$.
• References (p. 92, page image), four items: [A-K], [B], [Erd] and [M-P-S], as listed above.

Compiled scope

The paper is compiled at statement depth for its four results. Proposition 1.3, the result the citing problem consumes, with the definitions of sumfree and $S(B)$ and the formulation (2.1), read on the page images together with its proof, is paged on proposition_1_3. Proposition 1.4, Proposition 1.7 and the $\delta_k\to0$ statement (8.4) of § 8 are paged on proposition_1_4, proposition_1_7 and display_8_4, their statements read on the page images and their proofs for structure only. Nothing here is independently reviewed.

Bears on. #792: Proposition 1.3 (printed p. 72, PDF p. 2) is the bound $f(n)\ge(n+2)/3$ the site attributes to the paper, paged on proposition_1_3: "$S(B)\ge\frac13(|B|+2)$, for any $B\subset\mathbb Z_+$", where "$S(B)$ denotes the maximum size of a sumfree subset of $B$" and sumfree means $(A+A)\cap A=\emptyset$ (p. 71), the problem's convention that forbids $a+b=c$ with $a=b$ allowed. The proof (pp. 74--76) concludes (3.24) "for any $B\subset\mathbb Z_+$, $|B|\ge3$" (p. 76, PDF p. 6), the range under which Eberhard, Green and Manners quote it ("for $n\ge3$", p. 1), while Bedert states the bound with no size condition (p. 2); the set $\{1,2\}$ has $S=1<\frac43$, so the restriction is needed. The paper states the bound for positive integers, where Alon and Kleitman state theirs for nonzero integers. Proposition 1.4 (p. 72) is a lower bound for $S(B)$ whose excess over $|B|/3$ is $c_1\|\sum_{k\in B}\cos2\pi k\theta\|_1/\log|B|$; the paper draws from it no bound on $f(n)$ beyond Proposition 1.3, and it is the $L^1$ route that Bedert's paper develops into the $c\log\log n$ bound. Remark 1 of § 7 (p. 88, PDF p. 18, read on the page image) records why the asymmetric-interval method of Proposition 1.7 does not reach $S(B)$: any $\Omega\subset\mathbb T$ with $|\Omega|=\frac13$ and $(\Omega+\Omega)\cap\Omega=\emptyset$ satisfies $\Omega=\mathbb T\setminus(\Omega-\Omega)$ by Kneser's theorem and so is symmetric, which rules out the asymmetric two-arc device of § 7 for $S(B)$. The problem page reads Proposition 1.3 on the page image at statement depth and its proof for structure; no proof was checked.

Results.

• Proposition 1.3 (p. 72): $S(B)\ge\frac13(|B|+2)$ for $B\subset\mathbb Z_+$, proved for $|B|\ge3$ (p. 76). Bears on #792.
• Proposition 1.4 (p. 72; proof p. 77): $S(B)\ge\frac{|B|}3+c_1(\log|B|)^{-1} |\sum_{k\in B}\cos2\pi k\theta|_1$ with $c_1$ a fixed constant. Bears on #792 as a lower bound for $S(B)$ in terms of an $L^1$ norm.
• Proposition 1.7 (p. 72; proof pp. 87--88): $S_3(B)>\frac{|B|}4+c\frac{\log|B|}{\log\log|B|}$ for the largest $A\subset B$ with $(A+A+A)\cap A=\emptyset$. No Erdős problem in this corpus concerns $S_3$; its Remark 1 (p. 88) records why the method does not reach $S(B)$.
• Display (8.4) (p. 89; construction pp. 90--91): a bound $s_k(A)>\delta_k|A|$ valid for every finite $A\subset\mathbb Z_+$ forces $\delta_k\to0$ as $k\to\infty$, by examples resting on Lemma 8.5, stated without proof. No Erdős problem in this corpus concerns $s_k$ for $k\ge3$.

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