Almost primes in short intervals (Q5917782)

Let \(x\) be sufficiently large and \(\theta= {109\over 250}= 0.436\). The author proves that there are \(\gg {x^\theta\over \log x}\) integers with at most two prime factors in the interval \((x- x^\theta, x]\). This improves on earlier results of a similar type. The improvement comes from a better estimation of exponential sums of the form \(\sum_{h\sim H} \sum_{m\sim M} \sum_{n\sim N} a_m b_n e(hx/mn)\), where \(e(x)= e^{2\pi ix}\) and \(H\leq MNX^{- \theta+ \varepsilon}\).