Something like this?
events <- as.Date(c("2000-01-05", "2000-02-08", "2000-04-09", "2000-02-08",
"2000-03-13", "2000-03-13"))
week_vector <- seq(from = as.Date("2000-01-01"), to = as.Date("2000-12-23"), by = 7)
# or arguments more similar to the wording in the question, "52 [dates], 7 days periods":
week_vector <- seq(from = as.Date("2000-01-01"), length.out = 52, by = 7)
events2 <- cut(events, breaks = week_vector)
table(events2)
# 2000-01-01 2000-01-08 2000-01-15 2000-01-22 2000-01-29 2000-02-05 2000-02-12 2000-02-19
# 1 0 0 0 0 2 0 0
# 2000-02-26 2000-03-04 2000-03-11 2000-03-18 2000-03-25 2000-04-01 2000-04-08 2000-04-15
# 0 0 2 0 0 0 1 0
# 2000-04-22 2000-04-29 2000-05-06 2000-05-13 2000-05-20 2000-05-27 2000-06-03 2000-06-10
# 0 0 0 0 0 0 0 0
# 2000-06-17 2000-06-24 2000-07-01 2000-07-08 2000-07-15 2000-07-22 2000-07-29 2000-08-05
# 0 0 0 0 0 0 0 0
# 2000-08-12 2000-08-19 2000-08-26 2000-09-02 2000-09-09 2000-09-16 2000-09-23 2000-09-30
# 0 0 0 0 0 0 0 0
# 2000-10-07 2000-10-14 2000-10-21 2000-10-28 2000-11-04 2000-11-11 2000-11-18 2000-11-25
# 0 0 0 0 0 0 0 0
# 2000-12-02 2000-12-09 2000-12-16
# 0 0 0