For a finite abelian group G, the Erdős-Ginzburg-Ziv constant s(G) is the smallest s such that every sequence of s (not necessarily distinct) elements of G has a zero-sum subsequence of length exp(
Let us fix a prime p. The Erdős-Ginzburg-Ziv problem asks for the minimum integer s such that any collection of s points in the lattice Z n contains p points whose centroid is also a lattice point in Z n . For large n, this is essentially equivalent to asking for the maximum size of a subset of F n p without p distinct elements summing to zero. In this paper, we give a new upper bound for this problem for any fixed prime p ≥ 5 and large n. In particular, we prove that any subset of F n p without p distinct elements summing to zero has size at most Cp • 2 √ p n , where Cp is a constant only depending on p. For p and n going to infinity, our bound is of the form p (1/2)•(1+o(1))n , whereas all previously known upper bounds were of the form p (1−o(1))n (with p n being a trivial bound).Our proof uses the so-called multi-colored sum-free theorem which is a consequence of the Croot-Lev-Pach polynomial method. This method and its consequences were already applied by Naslund as well as by Fox and the author to prove bounds for the problem studied in this paper. However, using some key new ideas, we significantly improve their bounds.
Fix a graph H and some p ∈ (0, 1), and let XH be the number of copies of H in a random graph G(n, p). Random variables of this form have been intensively studied since the foundational work of Erdős and Rényi. There has been a great deal of progress over the years on the large-scale behaviour of XH , but the more challenging problem of understanding the small-ball probabilities has remained poorly understood until now. More precisely, how likely can it be that XH falls in some small interval or is equal to some particular value? In this paper we prove the almost-optimal result that if H is connected then for any x ∈ N we have Pr(XH = x) ≤ n 1−v(H)+o(1) . Our proof proceeds by iteratively breaking XH into different components which fluctuate at "different scales", and relies on a new anticoncentration inequality for random vectors that behave "almost linearly".
We prove several different anti-concentration inequalities for functions of independent Bernoulli-distributed random variables. First, motivated by a conjecture of Alon, Hefetz, Krivelevich and Tyomkyn, we prove some “Poisson-type” anti-concentration theorems that give bounds of the form 1/e + o(1) for the point probabilities of certain polynomials. Second, we prove an anti-concentration inequality for polynomials with nonnegative coefficients which extends the classical Erdős–Littlewood–Offord theorem and improves a theorem of Meka, Nguyen and Vu for polynomials of this type. As an application, we prove some new anti-concentration bounds for subgraph counts in random graphs.
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