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Let n 1<n 2< cdots be a lacunary sequence of integers and f in L^2([0,1]) . Estimate the growth of, for almost all alpha , sum 1 leq k leq N f( alpha n k ). For example, is it true that, for almost all…
Find the smallest h(d) such that the following holds. There exists a function f: mathbb N to -1,1 such that, for every d geq 1 , max P d left lvert sum n in P d f(n) right rvert leq h(d), where $P d…
Let N(k, ell) be the minimal N such that for any f: 1, ldots,N to -1,1 there must exist a k -term arithmetic progression P such that left lvert sum n in P f(n) right rvert geq ell. Find good up…
Let alpha 0 and n geq 1 . Let F(n, alpha) be the largest k such that there exists some 2-colouring of the edges of K n in which any induced subgraph H on at least k vertices contains more than $ alpha binom…
Let alpha in[0,1/2) and n,t geq 1 . Let F^ (t) (n, alpha) be the smallest m such that we can 2 -colour the edges of the complete t -uniform hypergraph on n vertices such that if X subseteq [n] with $ lvert …