Bosznay 1989 lower estimation non averaging sets

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theorem: Bosznay's theorem that the largest non-averaging subset of the first n integers has more than c n^{1/4} elements for all large n, with its one-page proof by the non-averaging set i q^3 + i(i+1)/2, i = 1, ..., q-1, below 2q^4; the lower bound of Problem 186's F(N) = N^{1/4+o(1)}.

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Á. P. Bosznay, On the lower estimation of non-averaging sets, Acta Math. Hungar. 53 (1989), no. 1--2, 155--157, DOI 10.1007/BF02170066 (the DOI is the publisher's; the printed head reads "Acta Math. Hung. 53 (1--2) (1989), 155--157"); the author at the Department of Mathematics, Faculty of Mechanical Engineering, Technical University of Budapest; received 14 August 1986 (p. 157). Cited as [Bo89] on the problem pages. The edition cited is the publisher's version of record at https://doi.org/10.1007/BF02170066; no preprint or repository copy is known here. Its four references (p. 157) are Abbott, On a conjecture of Erdős and Straus on non-averaging sets of integers, Proc. Fifth British Combinatorial Conference, Congressus Numerantium XV (1975), 1--4; Abbott, On the Erdős--Straus non-averaging set problem, Acta Math. Hungar. 47 (1986), 117--119; Erdős and Straus, Non-averaging sets II, Combinatorial Theory and its Applications, Vol. II, Colloq. Math. Soc. János Bolyai 4 (1970), 405--411; and Straus, Non-averaging sets, Proc. Sympos. Pure Math. XIX, Amer. Math. Soc. (1971), 215--222. None of the four is held.

The copy read for this card is the publisher's digitized scan of the printed article: 3 pages, printed pp. 155--157 = PDF pp. 1--3 (printed p. $n$ is PDF p. $n-154$), a 2005 scan (the copy's metadata names a TIFF source and a May 2005 creation date) with an OCR text layer that reads the prose and garbles every display (exponents, subscripts, fractions and the inequality signs come out as scattered characters). No notice is printed on the scan; the publisher's article page (https://link.springer.com/article/10.1007/BF02170066, read 2026-10-02) shows "© Akadémiai Kiadó" under Rights and permissions and names no open access or Creative Commons license, every other right reserved.

Read status: the whole paper was read on the page images of PDF pp. 1--3 on 2026-09-22. Claims checked for the definition of a non-averaging set and of $f(n)$, the recalled bounds of Straus, Erdős and Straus and Abbott, and the Theorem (p. 155), read clause by clause on the page image. The proof (pp. 155--156, one page) was read in full on the page images and its steps were followed at filing; the convexity assertion (2) is printed with a one-clause justification and no further argument. The references and the received date (p. 157) were read on the page image. Nothing here is independently reviewed.

Contents

• Introduction (p. 155, page image). Quoted: "A set $S$ of positive integers is called non-averaging if the arithmetic mean of two or more members of $S$ never belongs to $S$. Denote by $f(n)$ the cardinality of a largest non-averaging subset of $\{1,2,\ldots,n\}$." The recalled bounds, with $c_1,c_2,\ldots$ "positive absolute constants": Straus [4] "raising the problem of estimation of $f(n)$, proved that $f(n)>\exp(c_1\sqrt{\log n})$"; Erdős and Straus [3] "have the result $f(n)<c_2n^{2/3}$"; Abbott [1] "proved that $f(n)>c_3n^{1/10}$"; later, in [2], "be obtained $f(n)>c_5n^{1/5}$ for all $n$ and $f(n)>c_5n^{1/5}(\log\log n)^{2/5}$ for infinitely many $n$" (the constant $c_4$ is skipped in print, and "be obtained" is a misprint for "he obtained"). Then: "In this paper we show that $f(n)>c_6n^{1/4}$ improving a method of Abbott."
• The Theorem (p. 155, page image), quoted: "For some $c_6>0$ and all sufficiently large $n$ we have (1) $f(n)>c_6n^{1/4}$."
• Proof (pp. 155--156, page images). "Let $n=2q^4$, $q$ an integer. Without loss of generality, it is enough to show (1) only for such $n$'s." The points $(x_i,y_i)=(iq,\,i(i+1)/2)$ for $i=1,2,\ldots,q-1$ satisfy $x_i<q^2$, $y_i<q^2$, $x_1<x_2<\cdots<x_{q-1}$, and "lie on a convex curve (a parabole), thus they are non-averaging in a stronger sense", the displayed (2): for any $\lambda_1,\ldots,\lambda_k>0$ and indices $i_1,\ldots,i_k$ and $j$, with different numbers among $i_1,\ldots,i_k$, $\sum_l\lambda_l(x_{i_l},y_{i_l})/\sum_l\lambda_l\ne(x_j,y_j)$. The set is (3) $n_i=x_iq^2+y_i$ ($i=1,\ldots,q-1$), "different integers" with $n_i<q^4+q^2\le2q^4=n$. Indirectly, if $n_j=(n_{i_1}+\cdots+n_{i_k})/k$ with $i_1,\ldots,i_k$ different, the average is rewritten over $q$ indices by choosing $i_{k+1}=\cdots=i_q=j$; by (3), (4) $x_jq^2+y_j=\frac{x_{i_1}+\cdots+x_{i_q}}q\cdot q^2+\frac{y_{i_1}+\cdots+y_{i_q}}q$; the first quotient is an integer because every $x_i$ is a multiple of $q$, so by (4) the second is too; both are $<q^2$ because every $x_i,y_i<q^2$; hence $y_j=(y_{i_1}+\cdots+y_{i_q})/q$ and $x_j=(x_{i_1}+\cdots+x_{i_q})/q$, "and these equations contradict (2). The theorem is proved."
• References and received date (p. 157, page image), listed above.

Filing observations, not review verdicts. The set has $q-1$ elements below $2q^4$, so the reduction to $n=2q^4$ uses that $f$ is nondecreasing, which the paper leaves unsaid. The printed bound $n_i<q^4+q^2$ is what the argument needs; in fact $n_{q-1}=q^4-q^3+(q^2-q)/2<q^4$, so the set lies in $\{1,\ldots,q^4\}$, which is how Pham and Zakharov (p. 1) and Conlon, Fox and Pham (p. 4) report the construction, as $n_i=iq^3+i(i+1)/2$ in $[q^4]$; both forms agree with (3). The step from (4) to the two equations is the uniqueness of the base-$q^2$ digits of $x_jq^2+y_j$, and (2) is the strict convexity of the parabola through the points: a weighted average of points of the graph of a strictly convex function with at least two distinct abscissas lies strictly above the graph. Neither step is spelled out in print. The paper's definition of non-averaging (a mean of two or more members never belongs to $S$) coincides with the site's for Problem 186, as that page's Formulation paragraph records.

Compiled scope

The paper is compiled at statement depth for the result the citing problems consume: the Theorem (p. 155) with its construction (pp. 155--156), read on the page images and paged on theorem. The proof was read in full and followed at filing; nothing is independently reviewed.

Bears on. #186: the Theorem (printed p. 155, PDF p. 1), "For some $c_6>0$ and all sufficiently large $n$ we have (1) $f(n)>c_6n^{1/4}$", is the lower bound $F(N)\gg N^{1/4}$ the site credits to [Bo89], with $f(n)$ the problem's $F(N)$; the construction (3), $n_i=iq^3+i(i+1)/2$ for $i=1,\ldots,q-1$ (pp. 155--156), is the one the introductions of Pham and Zakharov and of Conlon, Fox and Pham reproduce. With Theorem 1 of Pham and Zakharov, $F(N)\le N^{1/4+o(1)}$, it gives $F(N)=N^{1/4+o(1)}$, the order of growth up to the $o(1)$ in the exponent. #131: the same Theorem is the $\alpha=\frac14$ that the bound of p. 128 of Erdős, Lev, Rauzy, Sándor and Sárközy feeds into Straus's transfer theorem, $f(n)\gg n^\alpha\Rightarrow Q(n)\gg n^{\alpha/(1+\alpha)}$, to obtain $F(N)\gg N^{1/5}$ for non-dividing sets; the paper itself does not mention non-dividing sets, and Straus's transfer theorem is not held.

Results.

• Theorem (p. 155): $f(n)>c_6n^{1/4}$ for some $c_6>0$ and all sufficiently large $n$, by the non-averaging set $\{iq^3+i(i+1)/2:1\le i\le q-1\}$ below $2q^4$ (pp. 155--156).

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