Influences and Decision Trees

A celebrated theorem of Friedgut says that every function f : {0, 1}n → {0, 1} can be approximated by a function g : {0, 1}n → {0, 1} with ‖f−g‖2 ≤ ǫ which depends only on eO(If/ǫ) variables where If is the sum of the influences of the variables of f . Dinur and Friedgut later showed that this statement also holds if we replace the discrete domain {0, 1}n with the continuous domain [0, 1]n, under the extra assumption that f is monotone. They conjectured that the condition of monotonicity is unnecessary and can be removed. We show that certain constant-depth decision trees provide counter-examples to Dinur-Friedgut conjecture. This suggests a reformulation of the conjecture in which the function g : [0, 1]n → {0, 1} instead of depending on a small number of variables has a decision tree of small depth. In fact we prove this reformulation by showing that the depth of the decision tree of g can be bounded by eO(If/ǫ 2). AMS Subject Classification: 06E30 28A35