Code:
********************************************************************************

isid person day
sort person day
save "orgdata" , replace

********************************************************************************
********************************************************************************

drop if inrange(day, 1, 59)

qui su day , meanonly

qui forvalues i = 1/`=ceil(r(max)/60)' {

    bysort person : gen byte d60 = sum(((day-day[1]) >=60))  == 1
    bysort person : drop if sum(d60)  == 0

    capture confirm variable d60
    
    if ( _rc  == 111 ) {
        
        continue , break
    }

    tempfile d60`i'
      
    preserve
    
        keep if d60
        keep person day
        save "`d60`i''" , replace
        
    restore
    
    local d60fls = "`d60fls'" + " " + regexr("`d60`i''", "^.*\\", "")
    
    capture drop d60
}

clear

foreach fn of local d60fls {
    
    append using `c(tmpdir)'/`fn'
}

qui merge 1:1 person day using "orgdata", gen(d60)  assert(match using)
gen byte wanted2 = ( day  == 0 ) | ( d60  == 3 )
sort person day
drop d60  

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list , sepby(person)
list if want != wanted2

********************************************************************************
Code:
. list , sepby(person)

     +-------------------------------+
     | person   day   want   wanted2 |
     |-------------------------------|
  1. |      1     0      1         1 |
  2. |      1    96      1         1 |
  3. |      1   109      0         0 |
  4. |      1   111      0         0 |
  5. |      1   115      0         0 |
  6. |      1   172      1         1 |
     |-------------------------------|
  7. |      2     0      1         1 |
  8. |      2    45      0         0 |
  9. |      2    90      1         1 |
 10. |      2   115      0         0 |
 11. |      2   145      1         0 |
     +-------------------------------+

. list if want != wanted2

     +-------------------------------+
     | person   day   want   wanted2 |
     |-------------------------------|
 11. |      2   145      1         0 |
     +-------------------------------+