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    On the Olson and the Strong Davenport constants

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    A subset SS of a finite abelian group, written additively, is called zero-sumfree if the sum of the elements of each non-empty subset of SS 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, pp-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 pp-groups of rank at most 22, paralleling and building on recent results on this problem for the Olson constant

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

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    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,CRA,B,C \subset \mathbb{R} be Borel sets with 0<dimHBdimHAdimHA0 < {\dim_{\mathrm{H}}} B \leq {\dim_{\mathrm{H}}} A {\dim_{\mathrm{H}}} A. Then, there exists cCc \in C such that dimH(A+cB)>dimHA.{\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 ss-Furstenberg set FR2F \subset \mathbb{R}^{2} has Hausdorff dimension dimHFmax{2s+(1s)2/(2s),1+s}. {\dim_{\mathrm{H}}} F \geq \max\{ 2s + (1 - s)^{2}/(2 - s), 1+s\}. We prove that every (s,t)(s,t)-Furstenberg set FR2F \subset \mathbb{R}^{2} associated with a tt-Ahlfors-regular line set has dimHFmin{s+t,3s+t2,s+1}.{\dim_{\mathrm{H}}} F \geq \min\left\{s + t,\tfrac{3s + t}{2},s + 1\right\}. Let πθ\pi_{\theta} denote projection onto the line spanned by θS1\theta\in S^1. We prove that if KR2K \subset \mathbb{R}^{2} is a Borel set with dimH(K)1{\dim_{\mathrm{H}}}(K)\le 1, then dimH{θS1:dimHπθ(K)<u}max{2(2udimHK),0}, {\dim_{\mathrm{H}}} \{\theta \in S^{1} : {\dim_{\mathrm{H}}} \pi_{\theta}(K) < u\} \leq \max\{ 2(2u - {\dim_{\mathrm{H}}} K),0\}, whenever udimHKu \leq {\dim_{\mathrm{H}}} K, and the factor "22" on the right-hand side can be omitted if KK is Ahlfors-regular.Comment: 71 pages. v3:corrected proof of Theorem 5.7, other small fixes, main results unchange
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