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For $\beta>1$ and $0<\eta<1$, there are constants $c>0$ and a positive integer $d$ with the following property in the needed domain $m\ge2$. If $A\subseteq[n]$, $|A|=m$, $n\le m^\beta$, and the real parameter $s$ satisfies

$$m^\eta\le s\le cm/\log m,$$

there exist a retained set $\widehat A\subseteq A$ of size at least $m-c^{-1}s\log m$, a proper GAP $P$ of dimension at most $d$ containing $\widehat A\cup\{0\}$, and a set $A'\subseteq\widehat A$ with $|A'|\le s$ whose subset-sum set $\Sigma(A')$ contains a homogeneous translate of the dilate $csP$, this dilate being proper.

A GAP has integer coordinates $\{x_0+\sum_{i=1}^k n_id_i:0\le n_i<L_i\}$. Properness means distinct coordinate tuples give distinct sums. Homogeneity means $\gcd(d_1,\ldots,d_k)\mid x_0$. In homogeneous coordinates $P=\{\sum_i n_id_i:a_i\le n_i\le b_i\}$, real dilation scales the coordinate endpoints, with coordinates still integral. For integer dilation it agrees with the repeated sumset.

This is the external Theorem 1.5 on p. 3 of the retained Conlon–Fox–Pham source, arXiv:2311.01416v1. Its original p. 2 definitions and p. 3 statement were checked; its full proof is not reconstructed here.

The original theorem and the Egyptian transcription both assert $A'\subseteq\widehat A$. These are distinct sets with distinct roles: the large retained set lies in the progression, while the small witness supplies its subset sums. The compiled application also converts centered signed representatives to a positive input before applying the theorem. The condition $m\ge2$ makes the displayed $\log m$ denominator defined; all applications have $m\to\infty$.

Source: published PDF, p. 7, Theorem 3; earlier v1 p. 5, Theorem 6.

Bears on. Problem 297.