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Statement

Definitions (printed p. 155): "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 paper's $c_1,c_2,\ldots$ "are positive absolute constants".

Theorem (printed p. 155, the paper's single theorem, unnumbered). "For some $c_6>0$ and all sufficiently large $n$ we have

$$f(n)>c_6n^{1/4}. \tag{1}$$

"

The introduction states it as "$f(n)>c_6n^{1/4}$ improving a method of Abbott", after recalling Abbott's $f(n)>c_5n^{1/5}$ for all $n$ and $f(n)>c_5n^{1/5}(\log\log n)^{2/5}$ for infinitely many $n$ (Acta Math. Hungar. 47 (1986), the paper's [2]).

The construction (pp. 155--156). For an integer $q$, the numbers

$$n_i=x_iq^2+y_i=iq^3+\frac{i(i+1)}2\qquad(i=1,\ldots,q-1), \tag{3}$$

where $(x_i,y_i)=(iq,\,i(i+1)/2)$, are $q-1$ distinct integers with $n_i<q^4+q^2\le2q^4$, and the set $\{n_1,\ldots,n_{q-1}\}$ is non-averaging. In fact $n_{q-1}=q^4-q^3+(q^2-q)/2<q^4$, so the set lies in $\{1,\ldots,q^4\}$, the form in which Pham and Zakharov (p. 1) and Conlon, Fox and Pham (p. 4) recall it; the paper's own bound is the cruder $2q^4$.

In the problem's notation. Problem 186's $F(N)$ is $f(N)$ (the definitions coincide, as that page's Formulation paragraph records), so the Theorem is $F(N)\gg N^{1/4}$. Problem 131's non-dividing sets are non-averaging, and Straus's transfer theorem turns $f(n)\gg n^{1/4}$ into $F(N)\gg N^{1/5}$ for that problem, a deduction made in the bound of p. 128 of Erdős, Lev, Rauzy, Sándor and Sárközy and not in this paper.

Source. 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 Theorem and the start of the proof on printed p. 155 (PDF p. 1 of the publisher's scan), the rest of the proof on printed p. 156 (PDF p. 2), read on the page images (the text layer garbles every display). The artifact is identified in the source digest.

Read depth. Claims checked: the definitions, the recalled bounds and the statement were read clause by clause on the page image. The proof (one page) was read in full on the page images and its steps were followed; the convexity assertion (2) is printed with a one-clause justification ("the points $(x_i,y_i)$ lie on a convex curve (a parabole)") and no further argument. Nothing here is independently reviewed.

Proof pointer

Pages 155--156. Take $n=2q^4$; "Without loss of generality, it is enough to show (1) only for such $n$'s" (the reduction uses that $f$ is nondecreasing, which is left unsaid). The points $(x_i,y_i)=(iq,\,i(i+1)/2)$, $i=1,\ldots,q-1$, have $x_i<q^2$, $y_i<q^2$ and increasing abscissas, and lie on a parabola, so they are "non-averaging in a stronger sense": (2) for any $\lambda_1,\ldots,\lambda_k>0$ and indices $i_1,\ldots,i_k$ and $j$ with different numbers among $i_1,\ldots,i_k$,

$$\frac{\sum_{l=1}^k\lambda_l(x_{i_l},y_{i_l})}{\sum_{l=1}^k\lambda_l}\ne(x_j,y_j).$$

Suppose $n_j=(n_{i_1}+\cdots+n_{i_k})/k$ with $i_1,\ldots,i_k$ different. Padding with $i_{k+1}=\cdots=i_q=j$ gives $n_j=(n_{i_1}+\cdots+n_{i_q})/q$, and by (3)

$$x_jq^2+y_j=\frac{x_{i_1}+\cdots+x_{i_q}}q\cdot q^2+\frac{y_{i_1}+\cdots+y_{i_q}}q. \tag{4}$$

The first quotient is an integer, since every $x_i$ is a multiple of $q$; by (4) the second quotient is then an integer too; both are $<q^2$, since every $x_i$ and $y_i$ is. "This and (4) imply" $y_j=(y_{i_1}+\cdots+y_{i_q})/q$ and $x_j=(x_{i_1}+\cdots+x_{i_q})/q$ (the uniqueness of the base-$q^2$ digits, unsaid in print), "and these equations contradict (2). The theorem is proved." Behind (2) is the strict convexity of the parabola: a weighted average of points on the graph of a strictly convex function with at least two distinct abscissas lies strictly above the graph, so it is not a point of the graph. Followed at filing; not independently reviewed.

Dependencies

None within the paper beyond the displayed (2), (3) and (4). Outside it, the recalled earlier bounds (Straus 1971, Erdős and Straus 1970, Abbott 1975 and 1986; the paper's [1]--[4], none held) are context and are not used in the proof.

Bears on

• Problem 186: the lower bound $F(N)\gg N^{1/4}$ that the site credits to [Bo89]; with Theorem 1 of Pham and Zakharov it gives $F(N)=N^{1/4+o(1)}$, the order of growth up to the $o(1)$ in the exponent.
• Problem 131: the $\alpha=\frac14$ input to Straus's transfer theorem in the bound of p. 128, which yields $F(N)\gg N^{1/5}$ for non-dividing sets; the transfer theorem itself is not held.