Arithmetic Progression-Free Subset-Sum Sets

For a finite set $A$ of positive integers, let $H(A)$ be its set of subset sums, including the empty sum, and let $g_k(n)$ be the least $N$ for which some $n$-element set $A\subseteq [N]$ has $H(A)$ free of nonconstant $k$-term arithmetic progressions. The problem of determining $g_k(n)$ was posed by Erdős and Sárkőzy. In the three-term case, we prove a lower bound equal to the exact bandwidth of the ternary grid. If $T_m=[x^m](1+x+x^2{)}^m$ is the central trinomial coefficient, then $$g_3(n)\ge \frac{T_n-1}{2}+\sum _{j=0}^{n-1}{T}_j=(\frac{\sqrt{3}}{2\sqrt{\pi }}+o(1))\frac{3^n}{\sqrt{n}}\mathrm{.}$$ For general $k\ge 4$ we show $$g_k(n){\gg }_k{\left(\frac{k-1}{k-2}\right)}^n{n}^{-{\log }_2((k-1)\mathrm{/}(k-2))}$$ In the opposite direction, a carry-free digit construction based on nearly-regular graphs gives $$\underset{n\to \mathrm{\infty }}{\mathrm{lim\hspace{0.17em}sup}}{g}_k(n{)}^{1\mathrm{/}n}\le \underset{p\mathrm{p}\mathrm{r}\mathrm{i}\mathrm{m}\mathrm{e},p\ge 3}{\min }{p}^{2\mathrm{/}(\min \{p,k\}-1)}\mathrm{.}$$ Consequently, as $k\to \mathrm{\infty }$, the logarithm of the lower exponential rate is at least $(1+o(1))\mathrm{/}k$, while the logarithm of the upper exponential rate is at most $(2+o(1))\log k\mathrm{/}k$.