On the Olson and the Strong Davenport constants

A subset $S$ of a finite abelian group, written additively, is called zero-sumfree if the sum of the elements of each non-empty subset of $S$ is non-zero. We investigate the maximal cardinality of zero-sumfree sets, i.e., the (small) Olson constant. We determine the maximal cardinality of such sets for several new types of groups; in particular, $p$-groups with large rank relative to the exponent, including all groups with exponent at most five. These results are derived as consequences of more general results, establishing new lower bounds for the cardinality of zero-sumfree sets for various types of groups. The quality of these bounds is explored via the treatment, which is computer-aided, of selected explicit examples. Moreover, we investigate a closely related notion, namely the maximal cardinality of minimal zero-sum sets, i.e., the Strong Davenport constant. In particular, we determine its value for elementary $p$-groups of rank at most $2$, paralleling and building on recent results on this problem for the Olson constant

Projections, Furstenberg sets, and the ABCABC sum-product problem

We make progress on several interrelated problems at the intersection of geometric measure theory, additive combinatorics and harmonic analysis: the discretised sum-product problem, exceptional estimates for orthogonal projections, and the dimension of Furstenberg sets. We give a new proof of the following asymmetric sum-product theorem: Let $A,B,C \subset \mathbb{R}$ be Borel sets with $0 < {\dim_{\mathrm{H}}} B \leq {\dim_{\mathrm{H}}} A {\dim_{\mathrm{H}}} A$. Then, there exists $c \in C$ such that $${\dim_{\mathrm{H}}} (A + cB) > {\dim_{\mathrm{H}}} A.$$ Here we only mention special cases of our results on projections and Furstenberg sets. We prove that every $s$-Furstenberg set $F \subset \mathbb{R}^{2}$ has Hausdorff dimension $${\dim_{\mathrm{H}}} F \geq \max\{ 2s + (1 - s)^{2}/(2 - s), 1+s\}.$$ We prove that every $(s,t)$-Furstenberg set $F \subset \mathbb{R}^{2}$ associated with a $t$-Ahlfors-regular line set has $${\dim_{\mathrm{H}}} F \geq \min\left\{s + t,\tfrac{3s + t}{2},s + 1\right\}.$$ Let $\pi_{\theta}$ denote projection onto the line spanned by $\theta\in S^1$. We prove that if $K \subset \mathbb{R}^{2}$ is a Borel set with ${\dim_{\mathrm{H}}}(K)\le 1$, then $${\dim_{\mathrm{H}}} \{\theta \in S^{1} : {\dim_{\mathrm{H}}} \pi_{\theta}(K) < u\} \leq \max\{ 2(2u - {\dim_{\mathrm{H}}} K),0\},$$ whenever $u \leq {\dim_{\mathrm{H}}} K$, and the factor "$2$" on the right-hand side can be omitted if $K$ is Ahlfors-regular.Comment: 71 pages. v3:corrected proof of Theorem 5.7, other small fixes, main results unchange