Lower Bounds for the Turán Densities of Daisies

For integers $r \geq 3$ and $t \geq 2$, an $r$-uniform {\em $t$-daisy} $\D^t_r$ is a family of $\binom{2t}{t}$ $r$-element sets of the form$$\{S \cup T \ : T\subset U, \ |T|=t \}$$for some sets $S,U$ with $|S|=r-t$, $|U|=2t$ and $S \cap U = \emptyset$. It was conjectured by Bollobás, Leader and Malvenuto (and independently by Bukh) that the Turán densities of $t$-daisies satisfy $\lim\limits_{r \to \infty} \pi(\D_r^t) = 0$ for all $t \geq 2$; this has become a well-known problem, and it is still open for all values of $t$. In this paper, we give lower bounds for the Turán densities of $r$-uniform $t$-daisies. To do so, we introduce (and make some progress on) the following natural problem in additive combinatorics: for integers $m \geq 2t \geq 4$, what is the maximum cardinality $g(m,t)$ of a subset $R$ of $\mathbb{Z}/m\mathbb{Z}$ such that for any $x \in \mathbb{Z}/m\mathbb{Z}$ and any $2t$-element subset $X$ of $\mathbb{Z}/m\mathbb{Z}$, there are $t$ distinct elements of $X$ whose sum is not in the translate $x+R$? This is a slice-analogue of an extremal Hilbert cube problem considered by Gunderson and Rődl as well as Cilleruelo and Tesoro.