How much are increasing sets positively correlated?
The author studies sets \(A\) of sequences consisting of 0 and 1 of length \(n\) (\(n\) a positive integer). Such a set \(A\) is called increasing (or upward directed) if for each sequence \(x\) in \(A\) every sequence \(y\) that is componentwise greater or equal to \(x\) is also in the set \(A\). He introduces the normalized counting measure \(\mu(A)=2^{-n}\text{card}(A)\), where \(\text{card}(A)\) is the cardinality of \(A\). If \(x\) is a sequence, the sequence \(T_i(x)\) is obtained from \(x\) by changing the \(i\)th coordinate. For a set \(A\) the set \(A_i\) is defined as \(A_i= \{x\) in \(A\mid T_i(x)\) not in \(A\}\). The main theorem states the following: If \(\varphi(x)\) is defined for \(0\leq x\leq 1\) as \(\varphi(x)=x/\log(e/x)\), then for some universal constant \(K\) and for all \(n\) and all increasing sets \(A\), \(B\) of sequences of 0 and 1 of length \(n\) we have that \[ \mu(A\cap B)- \mu(A)\mu(B)\geq 1/K^*\varphi\Biggl(\sum_{i\leq n}\mu(A_i)\mu(B_i)\Biggr). \] The author shows that this lower bound is more or less tight by giving an example from which it is clear that it is of the right order. The proof of the theorem is given starting from some relatively easy formulas followed by elaborate manipulations on them using Fubini's theorem and the Cauchy-Schwarz inequality several times finally reaching the aimed result.
scientific article; zbMATH DE number 936875