Lower bounds for the density variant of the Hilbert cube problem

Question

Given $\delta>0$ and positive integers $k$, write $h(\delta;k)$ for the smallest $N$ such that for any $S\subset [N]:=\{1,\dots,N\}$ of size $\ge \delta N$, there exists non-zero integers $n_0,d_1,\dots,d_k$ such that $n_0+\sum_{i\in I}d_i\in S$ for all $I\subset [k]$ (these sums need not all be distinct, in fact, they may be any $k$-term arithmetic progression).

It occurs to me that there is a short proof that for any fixed $\delta$, that $h(\delta;k)$ grows superexponentially as $k\to \infty$ (i.e., for any $C$, we have $h(\delta;k)>C^k$ for all large $k$).

Is such a bound recorded in the literature? I am aware of the work of Gunderson, Rodl, and Siderenko (e.g., "Extremal problems for sets forming Boolean algebras and complete partite hypergraphs"), but their bounds fixate on when $k$ is fixed and $\delta\to 0$.

Answer 1

If I read Theorem 1.6 of

Sándor, Csaba, Non-degenerate Hilbert cubes in random sets, J. Théor. Nombres Bordx. 19, No. 1, 249-261 (2007). ZBL1126.11014.

correctly, I think he shows (in your language) that for $\delta=1/2$ one has

$$ k \leq (1+\varepsilon)\log_2 \log_2 h(1/2;k)$$

for $k$ sufficiently large depending on $\varepsilon$, or equivalently $$ h(1/2;k) \gg_\varepsilon 2^{2^{k/(1+\varepsilon)}},$$

by computing the largest cube that can be located inside a random subset of $[n]$ (which has density at least $1/2$ at least half of the time). This is sharp up to the epsilon loss, as noted in that paper. In fact Sándor shows a more precise estimate with some lower order terms that I will not detail here.