A Lebesgue measurable set in the real line has Lebesgue density 0 or 1 at almost every point. Kolyada showed that there is a positive constant \(\delta\) such that for non-trivial measurable sets there is at least one point with upper and lower densities lying in the interval \((\delta, 1-\delta)\). Both Kolyada and later Szenes gave bounds for the largest possible value of this \(\delta\). In this note we reduce the best known upper bound, disproving a conjecture of Szenes.
"Points of Middle Density in the Real Line." Real Anal. Exchange 37 (2) 243 - 248, 2011/2012.