Postgres 13

By using db<>fiddle, you agree to license everything you submit by Creative Commons CC0.
CREATE TABLE tbl (date date, email text);
INSERT INTO tbl VALUES
  ('2012-01-01', 'test@test.com')
, ('2012-01-01', 'test1@test.com')
, ('2012-01-01', 'test2@test.com')
, ('2012-01-02', 'test1@test.com')
, ('2012-01-02', 'test2@test.com')
-- , ('2012-01-03', 'test@test.com')  -- no entry for this day
, ('2012-01-04', 'test@test.com')
, ('2012-01-05', 'test@test.com')
, ('2012-01-05', 'test@test.com')
, ('2012-01-06', 'test@test.com')
, ('2012-01-06', 'test@test.com')
, ('2012-01-06', 'test1@test.com`')
;

CREATE TABLE



INSERT 0 11


-- only days with entry
SELECT date
     ,(SELECT count(DISTINCT email)
       FROM   tbl
       WHERE  date BETWEEN t.date - 2 AND t.date -- period of 3 days
      ) AS dist_emails
FROM   tbl t
WHERE  date BETWEEN '2012-01-01' AND '2012-01-06'  
GROUP  BY 1
ORDER  BY 1;


date
dist_emails


2012-01-01
3

2012-01-02
3

2012-01-04
3

2012-01-05
1

2012-01-06
2



SELECT 5


-- all days
SELECT date
     ,(SELECT count(DISTINCT email)
       FROM   tbl
       WHERE  date BETWEEN g.date - 2 AND g.date
      ) AS dist_emails
FROM  (SELECT generate_series(timestamp '2012-01-01'
                            , timestamp '2012-01-06'
                            , interval  '1 day')::date) AS g(date);


date
dist_emails


2012-01-01
3

2012-01-02
3

2012-01-03
3

2012-01-04
3

2012-01-05
1

2012-01-06
2



SELECT 6


SELECT g.date, count(DISTINCT email) AS dist_emails
FROM  (SELECT generate_series(timestamp '2012-01-01'
                            , timestamp '2012-01-06'
                            , interval  '1 day')::date) AS g(date)
LEFT   JOIN tbl t ON t.date BETWEEN g.date - 2 AND g.date
GROUP  BY 1
ORDER  BY 1;


date
dist_emails


2012-01-01
3

2012-01-02
3

2012-01-03
3

2012-01-04
3

2012-01-05
1

2012-01-06
2



SELECT 6
date	dist_emails
2012-01-01	3
2012-01-02	3
2012-01-03	3
2012-01-04	3
2012-01-05	1
2012-01-06	2