Firebase let authenticated user create their own user entry - swift

Question

I wanted to know if it's possible that a newly authenticated user can insert only their information to the realtime database.
The flow would be something like:
User authenticates via my app where he has to input
Email
Username
...
The user is authenticated (via Firebase Authentication) and has a uid
User writes a new DB entry in the /users/{uid} field of his/her. However, they are not allowed to set the balances field below.
Email
Username
...
He/She should only be able to write and read his/her own /users/{uid} object.
I've tried these rules so far, but none of them lets me create a new /users/{uid}
{
"rules": {
"users": {
"$uid": {
".read": "$uid === auth.uid",
"balances": {
".write": false
},
"userName": {
".write": "$uid === auth.uid",
},
"email": {
".write": "$uid === auth.uid",
},
"phoneNumber": {
".write": "$uid === auth.uid",
}
}
}
}
}
The code I want to insert:
// ...
Auth.auth().createUser(withEmail: email, password: password) { authResult, error in
if let error = error {
self.error = error.localizedDescription
return
}
if let authResult = authResult {
// Auth successful
User.firebaseUser = authResult.user
print("Firebace UserId: \(authResult.user.uid)")
let ref: DatabaseReference! = Database.database().reference()
ref.child("users").child(authResult.user.uid).setValue([
"userName": self.userName,
"email": self.email
])
// ...
}
}
My Swift error message:
2022-05-26 16:55:55.043864+0200 Project[15240:1265346] 9.0.0 - [FirebaseDatabase][I-RDB038012] setValue: or removeValue: at /users/EZ2bNl3rMANtLtn1G9Wfqi0Ivjl1 failed: permission_denied
Thank you in advance!