Ok, hope this is the right place for this.

I was working on my guitar last night trying to get the intonation really tight and I thought I got everything set up perfectly. However, I noticed that I am still sharp on the first three frets of most of the strings. On the rest, it is spot on. What might be causing this? Does the nut need to be slotted a little deeper? And if so, can I do this myself? I have an Ibanez AS73B.

Also, I have the saddle all of the back on my low E string and I am still a little sharp. Anything I can do about that?

I appreciate all of your help.
turn the saddle around
tone slut

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Sounds like you're not performing the adjustments correctly.
You should check the intonation with the guitar in playing position, not flat on it's back.
Never touch the tuners while checking the pitch as this can skew the tone slightly.
I prefer checking open string vs fretted twelth note (octave) as it is supposed to compensate for the slight string length change that occurs when you depress the note.
If your nut is seriously too high it could certainly throw off the first few frets a bit but that's pretty rare, the 1st, 2nd & 3rd frets would be hard to play in that case and nut height is often lower for easier fretting on the low frets.

As for the saddle all the way back, if it's a tune o matic you can reverse the saddle so the ridge is closer to the back (if it isn't already) to give more adjustment play.
Moving on.....
Ok. I was checking it in playing position the whole time. What do you mean by checking the open string rather than fretted twelfth note? How do you check it by open string?
I mean compare the Open Note and the Twelth Fretted note not the harmonic. Sorry about the confusing verbage!
Moving on.....
Oh I see. That was actually what I was doing. For the same reason. I felt it would compensate for the stretch. But, like I said, still sharp on the first three. I'll try turning the saddle around on the low E to get that one a little better.

Anything else I can do about the first three?
Can you measure the distance between the bottom of the strings and the crown of the first fret?
Also, tell me a little about your guitar, what make, anything special done to it etc?
Moving on.....
I'll try and measure the distance tonight when I get home. It is a 2009 Ibanez AS73 semi-hollow in brown sunburst. All stock parts. All I've done it adjust the intonation, lower the action, and adjust the pickup height. What should the measurement be?

Also, I checked the saddle and they are all already flipped so it is as far back as it can go.