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We study analogues of Sidorenko’s conjecture and the forcing conjecture in oriented graphs, showing that natural variants of these conjectures in directed graphs are equivalent to the asymmetric, undirected analogues of the conjectures. 

Abstract For a subset$$A$$of an abelian group$$G$$, given its size$$|A|$$, its doubling$$\kappa =|A+A|/|A|$$, and a parameter$$s$$which is small compared to$$|A|$$, we study the size of the largest sumset$$A+A'$$that can be guaranteed for a subset$$A'$$of$$A$$of size at most$$s$$. We show that a subset$$A'\subseteq A$$of size at most$$s$$can be found so that$$|A+A'| = \Omega (\!\min\! (\kappa ^{1/3},s)|A|)$$. Thus, a sumset significantly larger than the Cauchy–Davenport bound can be guaranteed by a bounded size subset assuming that the doubling$$\kappa$$is large. Building up on the same ideas, we resolve a conjecture of Bollobás, Leader and Tiba that for subsets$$A,B$$of$$\mathbb{F}_p$$of size at most$$\alpha p$$for an appropriate constant$$\alpha \gt 0$$, one only needs three elements$$b_1,b_2,b_3\in B$$to guarantee$$|A+\{b_1,b_2,b_3\}|\ge |A|+|B|-1$$. Allowing the use of larger subsets$$A'$$, we show that for sets$$A$$of bounded doubling, one only needs a subset$$A'$$with$$o(|A|)$$elements to guarantee that$$A+A'=A+A$$. We also address another conjecture and a question raised by Bollobás, Leader and Tiba on high-dimensional analogues and sets whose sumset cannot be saturated by a bounded size subset. 

Celebrated theorems of Roth and of Matoušek and Spencer together show that the discrepancy of arithmetic progressions in the first $$n$$ positive integers is $$\Theta (n^{1/4})$$ . We study the analogous problem in the $$\mathbb {Z}_n$$ setting. We asymptotically determine the logarithm of the discrepancy of arithmetic progressions in $$\mathbb {Z}_n$$ for all positive integer $$n$$ . We further determine up to a constant factor the discrepancy of arithmetic progressions in $$\mathbb {Z}_n$$ for many $$n$$ . For example, if $n=p^k$ is a prime power, then the discrepancy of arithmetic progressions in $$\mathbb {Z}_n$$ is $$\Theta (n^{1/3+r_k/(6k)})$$ , where $$r_k \in \{0,1,2\}$$ is the remainder when $$k$$ is divided by $$3$$ . This solves a problem of Hebbinghaus and Srivastav.