Fill gaps in SQL Server dates ranges

You can solve this using various techniques. In the example below, I am using an approach as the code is a body of SQL routine.

So, here is your date:

DECLARE @Periods TABLE
(
    [PeriodId] INT
   ,[PeriodStart] DATE
   ,[PeriodEnd] DATE
);

INSERT INTO @Periods ([PeriodId], [PeriodStart], [PeriodEnd])
VALUES ('202005', '2020-05-01', '2020-05-31')
      ,('202006', '2020-06-01', '2020-06-30');

DECLARE @Activities  TABLE
(
    [ActivityId] INT
   ,[UserId] CHAR(1)
   ,[ActivityStart] DATE
   ,[ActivityEnd] DATE
);

INSERT INTO @Activities ([ActivityId], [UserId], [ActivityStart], [ActivityEnd])
VALUES (1, 'A', '2020-05-20', '2020-06-05')
      ,(2, 'A', '2020-06-15', '2020-06-18')
      ,(3, 'B', '2020-06-10', '2020-06-25');

Then, let's say we have an input parameter @PeriodID and via it we are extracting the corresponding start and end date periods:

DECLARE @PeriodID INT
       ,@PeriodDateStart DATE
       ,@PeriodDateEnd DATE;

SET @PeriodID = 202006;

SELECT @PeriodDateStart = [PeriodStart]
      ,@PeriodDateEnd = [PeriodEnd]
FROM @Periods 
WHERE [PeriodId] = @PeriodID;

Then, let's create a buffer table in which we will calculated the matches between the activity and the period table and add start and end periods records if needed:

DECLARE @Buffer TABLE
(
    [ActivityId] INT
   ,[UserId] CHAR(1)
   ,[ActivityStart] DATE
   ,[ActivityEnd] DATE
);

WITH DataSource AS
(
    SELECT A.[ActivityId]
          ,A.[UserId]
          ,A.[ActivityStart]
          ,A.[ActivityEnd]
    FROM @Activities A
    INNER JOIN @Periods P
        ON A.[ActivityStart] <= P.[PeriodEnd]
        AND A.[ActivityEnd] >= P.[PeriodStart]
    WHERE P.PeriodId = @PeriodID
)
INSERT INTO @Buffer ([ActivityId], [UserId], [ActivityStart], [ActivityEnd])
SELECT [ActivityId]
      ,[UserId]
      ,IIF([ActivityStart] < @PeriodDateStart, @PeriodDateStart, [ActivityStart]) AS [ActivityStart]
      ,[ActivityEnd]
FROM DataSource 
UNION ALL
SELECT NULL
      ,[UserId]
      ,DATEADD(DAY, 1, MAX([ActivityEnd]))
      ,@PeriodDateEnd
FROM DataSource
GROUP BY [UserId]
HAVING DATEADD(DAY, 1, MAX([ActivityEnd])) < @PeriodDateEnd
UNION ALL
SELECT NULL
      ,[UserId]
      ,@PeriodDateStart
      ,DATEADD(DAY, -1, MIN([ActivityStart]))
FROM DataSource
GROUP BY [UserId]
HAVING DATEADD(DAY, -1, MIN([ActivityStart])) > @PeriodDateStart;

It's simple. In the common table expression I have used your code. And then, we just simply check if we need to add a record before or/and after the period for a specific user.

Now, we are ready to calculate the gaps, right? A lot of variants here. I am using the LEAD function in order to calculate the missing period for each row. The statement is below:

SELECT *
      ,DATEADD(DAY, 1, [ActivityEnd]) AS [MissingPeriodStart]
      ,DATEADD(DAY, -1, LEAD([ActivityStart]) OVER (PARTITION BY [UserID] ORDER BY [ActivityStart] ASC)) AS [MissingPeriodEnd]
FROM @Buffer
ORDER BY USERID, ActivityStart;

The output is like this:

So, you may see how we have generated missing periods dates for each row, except the last one. Now, we need to get only some of these missing periods. It's like this:

WITH DataSource AS
(
    SELECT *
          ,DATEADD(DAY, 1, [ActivityEnd]) AS [MissingPeriodStart]
          ,DATEADD(DAY, -1, LEAD([ActivityStart]) OVER (PARTITION BY [UserID] ORDER BY [ActivityStart] ASC)) AS [MissingPeriodEnd]
    FROM @Buffer
)
SELECT @PeriodID AS [PeriodID]
      ,[UserId]
      ,[ActivityId]
      ,[ActivityStart]
      ,[ActivityEnd]
FROM DataSource
UNION ALL 
SELECT @PeriodID AS [PeriodID]
      ,[UserId]
      ,NULL
      ,[MissingPeriodStart]
      ,[MissingPeriodEnd]
FROM DataSource
WHERE NOT EXISTS 
(
    SELECT 1 
    FROM DataSource DS
    WHERE [MissingPeriodStart] = DS.[ActivityStart]
        AND [UserID] = DS.[UserID]
)
    AND [MissingPeriodStart] < [MissingPeriodEnd]
ORDER BY [UserId]
        ,[ActivityStart];

and the result is:

Of course, this is an idea. You may need to change it or tune it in order to be used with your real data. I hope it will give you a start.

This is not the craziest gaps problem I've seen, but it's a good one.

DECLARE @PeriodId int = 202006;

DECLARE @ps date, @pe date;
SELECT @ps = PeriodStart, @pe = PeriodEnd FROM dbo.Periods
   WHERE PeriodId = @PeriodId;
   
;WITH dates(rn,dt) AS 
(
    SELECT 1, @ps UNION ALL SELECT rn + 1, DATEADD(DAY, rn, @ps) 
    FROM dates WHERE dt < @pe
)
groups(UserId, dt, ActivityId, grp) AS
(
  SELECT u.UserId, d.dt, r.ActivityId, 
    d.rn - DENSE_RANK() OVER (PARTITION BY u.UserId, r.ActivityStart ORDER BY d.dt)
  FROM dates AS d CROSS JOIN (SELECT DISTINCT UserId FROM dbo.Activities 
    WHERE @pe >= ActivityStart AND @ps <= ActivityEnd) AS u
  LEFT OUTER JOIN dbo.Activities AS r
  ON u.UserId = r.UserId AND d.dt >= r.ActivityStart AND d.dt <= r.ActivityEnd
)
SELECT PeriodId = @PeriodId, UserId, ActivityId,
  NewActivityStart = MIN(dt),
  NewActivityEnd   = MAX(dt)
FROM groups 
GROUP BY UserId, ActivityId, grp
ORDER BY UserId, NewActivityStart;

If a period can be over 100 days, you need MAXRECURSION at the end:

OPTION (MAXRECURSION 32767);  

If a period can be more than 32,767 days, change 32767 to 0.

Updated fiddle here.

I don't think this is that complicated. If you expand the periods into individual dates and do a left join, then this becomes a gaps-and-islands problem:

with dates as (
      select periodid, periodstart as dte, periodend
      from periods
      union all
      select periodid, dateadd(day, 1, dte), periodend
      from dates
      where dte < periodend
     )
select userid, activityid, min(dte), max(dte)
from (select d.dte, d.periodid, u.userid, a.activityid,
             row_number() over (partition by u.userid, a.activityid order by d.dte) as seqnum
      from dates d cross join
           (select distinct userid from activities) u left join
           activities a
           on a.userid = u.userid and
              a.activitystart <= d.dte and a.activityend >= d.dte
     ) da
group by userid, activityid, periodid, dateadd(day, -seqnum, dte)
order by userid, min(dte);

Here is a db<>fiddle.

Note: This produces results for all users and all periods -- which seems reasonable given your description. It is pretty simple to modify to filter out users with no activity during a given period.

Also, this does not go to the end of the month. Instead, it includes the complete periods. I don't see why months would play into this -- except to confuse matters -- consider if two periods have days in the same month, for instance.