60 days vs 4 days
This is only 15 periods. 1/15 = 0.06666666. This is FAR greater than your tolerance of 0.001.
In other words, you must test very often because you cannot afford (according to your tolerance) to miss even one instance.
All year 365 days vs 4 days is 91.25 perioods, giving 0.011, still much greater than your hoped-for tolerance.
1) You may wish to rethink your tolerance.
2) How many subjects?
3) Draw a picture.
10 days vs 4 days
On any day you test, you will catch only the users on that day and those who used 1, 2, or 3 days before the testing. This is part of your difficulty. With only a 10-day test, using can occur on any of 13 different days! It doesn't matter which day you pick, you will cover only 4 days of users if you do only one day. 4/13 = 30.77% - Again, nowhere near your 0.1%.
Pick another day, 4 days away, of course, and increase by 4. So, 1, 5, 9 will get 12/13 = 92.31%.
60 days, then, has 1, 5, 9, 13, 17, 21, 25, 29, 33, 37, 41, 45, 49, 53, 57 - 15 testing days.
Since 60 is a little irritating for this arrangement, you miss quite a few at the end. Picking 58 instead for 57 won't make any difference. You'll just get a different 4. In any case, you cover (15*4)/63 = 95.24%.
However, if you extended your study period to day 61, the results certainly are improved. You add a day, but you also add a testing day, giving (16*4)/64 = 100%
Pretty irritating, isn't it? You cannot make small adjustments.
Note: This considers only one (1) subject. As your population grows, adjustments will get smaller and it will become more tractable. When will you be able to make an adjustment so small as 0.1%? Good thought question.