If you want to get some fresh air in your home, do not leave a window half-open all day long—instead, open all your windows at once to cycle the air as fast as possible, then close them again and let the purifier do its job. If you do that in the morning (ideally just before leaving, if you aren't staying home), you'll have great air quality through the entire evening and night.
Since we're on HN, I'll also mention a toy project I built a while back [0]. It uses a Raspberry Pi to read particulate matter sensor mesurements over serial link from a Dylos 1100 Pro, then pipes it into Redis and InfluxDB and ultimately Grafana for presentation. I've used it to track the air quality in my home in for the past year or so and it's pretty fun. In my case, cooking is usually the biggest contributor to poor indoor air quality!