Given a set $S \subseteq \mathbb{R}^2$, define the \emph{Helly number of $S$}, denoted by $H(S)$, as the smallest positive integer $N$, if it exists, for which the following statement is true: for any finite family $\mathcal{F}$ of convex sets in~$\mathbb{R}^2$ such that the intersection of any $N$ or fewer members of~$\mathcal{F}$ contains at least one point of $S$, there is a point of $S$ common to all members of $\mathcal{F}$. We prove that the Helly numbers of \emph{exponential lattices} $\{α^n \colon n \in \mathbb{N}_0\}^2$ are finite for every $α>1$ and we determine their exact values in some instances. In particular, we obtain $H(\{2^n \colon n \in \mathbb{N}_0\}^2)=5$, solving a problem posed by Dillon (2021). For real numbers $α, β> 1$, we also fully characterize exponential lattices $L(α,β) = \{α^n \colon n \in \mathbb{N}_0\} \times \{β^n \colon n \in \mathbb{N}_0\}$ with finite Helly numbers by showing that $H(L(α,β))$ is finite if and only if $\log_α(β)$ is rational.

Abstract. We consider the minimum number of lines [Formula: see text] and [Formula: see text] needed to intersect or pierce, respectively, all the cells of the [Formula: see text] chessboard. Determining these values can also be interpreted as a strengthening of the classical plank problem for integer points. Using the symmetric plank theorem of K. Ball, we prove that [Formula: see text] for each [Formula: see text]. Studying the piercing problem, we show that [Formula: see text] for [Formula: see text], where the upper bound is conjectured to be sharp. The lower bound is proven by using the linear programming method, whose limitations are also demonstrated.

Abstract We improve the best known upper bound on the density of a planar measurable set A containing no two points at unit distance to 0.25442. We use a combination of Fourier analytic and linear programming methods to obtain the result. The estimate is achieved by means of obtaining new linear constraints on the autocorrelation function of A utilizing triple-order correlations in A, a concept that has not been previously studied.