When Theodore is on the stairs wearing his yellow shirt, the same girl passes him four times.
At the beginning of the movie, when Theodore was passing near the working woman, the computer in front of her, has no cables. After Theodore passes, the computer has cables plugged in.
When Theodore's ex-wife signs the divorce papers, the pen she uses changes between shots.
When Theodore finds his book in his mailbox, it fits the mailbox's size. When he reads it in his office, the book's size could not fit in the mailbox.
When Theodore is walking through the plaza, from his view we see an older Asian man is just about to pass him on his right, but then in the next shot from the other direction, no such man passes him.
On the spine of the physics book he was reading, "Unknown" was misspelled as "Unkown".
When Theodore is moving his forefingers to make the alien boy 3D video game run up the snow hill, it is not synchronized with the character's footsteps.
When Theodore is dictating the first letter, the computer program knows when to indent for the complimentary close, even though he gives no indication that that is what he is dictating.
When Theodore is lying in the couch at Amy's house there is a crew member behind Amy in the shadow.
The movie is set in Los Angeles. It is filmed in California and China. However, in some of the panoramic shots of the city the traffic is driving on the left side of the road. In both the US and in China, traffic drives on the right side of the road.
Writing letters for people is the work of a LLM. It seems odd the film missed the obvious use of the AI, to write letters in the manner of current LLM's. The actor even uses the AI to help him with the letters, spellcheck and grammar. It is unfortunate, but this is a glaring plot hole in the movie just 10yrs later.