Korsky 2026 arithmetic progression free subset sum sets

../

corollary_4_2: The integer-linear formulation of the three-term case: subset sums free of nonconstant three-term progressions are the same as injectivity of the linear form on {0,1,2}^n, so g_3(n) is a layout minimum over positive integer vectors; the characterization later preprints build on.

theorem_1_1: An exact finite lower bound for the least N such that some n-element subset of [N] has three-term-progression-free subset sums, in terms of central trinomial coefficients, with the asymptotic 3^n / sqrt(n) form.

theorem_1_2: The general-k lower bound for the least N whose n-element subsets can have k-term-progression-free subset sums, with exponential base (k-1)/(k-2) from a chain-expansion argument and an averaging step over unused generators.

theorem_1_3: The general-k upper bound for the least N whose n-element subsets can have k-term-progression-free subset sums, from a carry-free base-p digit construction with two-coordinate generators indexed by the edges of a nearly regular graph; with Corollary 1.4 on the large-k rates.

---

Samuel Korsky, Arithmetic Progression-Free Subset-Sum Sets. arXiv preprint (2026). arXiv:2606.24139v1 (23 June 2026), doi:10.48550/arXiv.2606.24139.

For a finite set $A$ of positive integers, $H(A)$ collects the sums of all subsets of $A$, the empty subset (sum $0$) included, and $g_k(n)$ is the least $N$ such that $[N]$ contains an $n$-element set $A$ whose $H(A)$ has no nonconstant $k$-term arithmetic progression (Section 1, p. 1), the function of Erdős and Sárközy behind Problem 817; the paper records their bound $g_3(n)\gg3^n/n^{O(1)}$ and their question whether $g_3(n)\gg3^n$ as "open in that form" (p. 2). Theorem 1.1 (p. 2) proves $g_3(n)\ge b_n:=(T_n-1)/2+\sum_{j<n}T_j$, with $T_m$ the $m$th central trinomial coefficient, hence $g_3(n)\ge(\sqrt3/(2\sqrt\pi)+o(1))3^n/\sqrt n$, through the characterization of Proposition 4.1 and Corollary 4.2 (p. 6: $H(A)$ is three-term-progression-free exactly when the $3^n$ ternary sums $\sum\varepsilon_ia_i$, $\varepsilon_i\in\{0,1,2\}$, are distinct, so $g_3(n)$ is a layout minimum on $\{0,1,2\}^n$) and the exact bandwidth of the ternary grid (Billera and Blanco); Remark 4.6 (p. 8) tabulates $g_3(n)=1,3,8,22$ for $n\le4$ against $b_n=1,3,8,21$ and notes the elementary $g_3(n)\le3^{n-1}$. Theorem 1.2 (p. 2) gives, for fixed $k\ge4$, $g_k(n)\gg_k((k-1)/(k-2))^nn^{-\log_2((k-1)/(k-2))}$ by a chain-expansion and averaging argument (Corollary 5.2, Theorem 5.4 and Corollary 5.5, pp. 9--10), improving the base $k/(k-1)$ of Dietmann and Elsholtz; Theorem 1.3 (p. 3) gives $g_k(n)<2p^{\rho_{p,k}(n)-1}$ for every prime $p\ge3$, hence $\limsup g_k(n)^{1/n}\le\min_pp^{2/(\min\{p,k\}-1)}$, by a carry-free base-$p$ digit construction with one two-coordinate generator for each edge of a nearly regular graph (Theorem 6.3, p. 12); Corollary 1.4 (p. 3) places the logarithms of the lower and upper exponential rates between $(1+o(1))/k$ and $(2+o(1))\log k/k$. Section 2 surveys related work (Erdős and Sárközy's 1992 paper, Hilbert cubes, bounded-coefficient dissociated sets).

The retained folder-name PDF is arXiv:2606.24139v1 (23 June 2026, 15 pp.; dated June 22, 2026 in its header); no journal record was found (Crossref bibliographic query, 2026-09-18): a preprint. Read status: claims checked for the definitions, Theorems 1.1--1.3, Corollary 1.4, Proposition 4.1, Corollary 4.2 and Remark 4.6 (pp. 1--3, 6, 8, text layer) on 2026-09-18; the proofs of Sections 4--6 were read for structure or for their statement labels only. Result pages: theorem_1_1, theorem_1_2, theorem_1_3 and corollary_4_2. The arXiv record (https://arxiv.org/abs/2606.24139, read 2026-10-02) names the Creative Commons Attribution 4.0 license.

Source: https://arxiv.org/abs/2606.24139.

Bears on. #817