TY - JOUR AB - Consider the sum X(ξ)=∑ni=1aiξi, where a=(ai)ni=1 is a sequence of non-zero reals and ξ=(ξi)ni=1 is a sequence of i.i.d. Rademacher random variables (that is, Pr[ξi=1]=Pr[ξi=−1]=1/2). The classical Littlewood-Offord problem asks for the best possible upper bound on the concentration probabilities Pr[X=x]. In this paper we study a resilience version of the Littlewood-Offord problem: how many of the ξi is an adversary typically allowed to change without being able to force concentration on a particular value? We solve this problem asymptotically, and present a few interesting open problems. AU - Bandeira, Afonso S. AU - Ferber, Asaf AU - Kwan, Matthew Alan ID - 9574 JF - Electronic Notes in Discrete Mathematics SN - 1571-0653 TI - Resilience for the Littlewood-Offord problem VL - 61 ER - TY - JOUR AB - We give several results showing that different discrete structures typically gain certain spanning substructures (in particular, Hamilton cycles) after a modest random perturbation. First, we prove that adding linearly many random edges to a dense k-uniform hypergraph ensures the (asymptotically almost sure) existence of a perfect matching or a loose Hamilton cycle. The proof involves an interesting application of Szemerédi's Regularity Lemma, which might be independently useful. We next prove that digraphs with certain strong expansion properties are pancyclic, and use this to show that adding a linear number of random edges typically makes a dense digraph pancyclic. Finally, we prove that perturbing a certain (minimum-degree-dependent) number of random edges in a tournament typically ensures the existence of multiple edge-disjoint Hamilton cycles. All our results are tight. AU - Krivelevich, Michael AU - Kwan, Matthew Alan AU - Sudakov, Benny ID - 9575 JF - Electronic Notes in Discrete Mathematics SN - 1571-0653 TI - Cycles and matchings in randomly perturbed digraphs and hypergraphs VL - 49 ER -