it's not the same, since synchronization needs you to have mobile's synchronization open all the time,
which is not required normally,
on the contrary, yahoo's app pushes the mails, irrespectively of what you do with phone's sync options!
Oh, man!
I think it is good to challenge people especially when they give "canned" responses. But if you are calling someone out on being wrong, it behooves to do your homework first and double check your information/knowledge.
Back to the issue of Push.
First, some terminology has to be defined: Push (in particular notifications for new e-mails) mean that it is the server that initiates the contact (hence the push). This is opposite to the mail client (app) initiating the contact ("synchronization" in terms of Aquamail - either scheduled or manual) -- "Pull".
Push can be implemented via variety of methods. You are right,for the server to contact the phone, you have to have a
connection (not
synchronization as you wrote) open. For that, there are two possibilities: (1) either the phone must maintain a persistent connection, or (2) the phone can have some sort of a "beacon", opening a connection every so often, and giving the server a windows of opportunities to communicate (via some other protocol), and once the server passes the message that there is new mail, the client can collect that new mail.
Several years ago, "Push" was implemented for IMAP4 protocol (standard) as an extension that is call "IMAP IDLE". It essentially requires the mail client (app) to maintain the connection open all the time.
Gmail own app does not use IMAP for connecting to Gmail server. And, correspondingly, it doesn't use IMAP IDLE for Push. Instead, it uses Google's proprietary protocol. (Google actually uses the so-called "Cloud" that is essentially a "holding" area, a buffer for different types of notifications that are awaiting to be transmitted for the apps that are "subscribed" for those notifications. As far as I understand, those notifications could be either very short notifications about new messages waiting, or maybe even the messages themselves (or their portions). I don't remember and I don't think it matters much for the present discussion.)
Gmail uses its own, specially crafted protocol for the communication between their server and their client (while also allowing other clients connecting to Gmail server via standard IMAP4 protocol, and the Gmail app (client) connecting to other IMAP servers). As a result, - they just use some optimized communication that is essentially a Push. Note, that Gmail app does not have (at least for Gmail accounts, I haven't used it for any other) the setting for the frequency of syncing.
As an aside: since Gmail uses its own, specially crafted protocol for the communication between the server and the client, - and Android is also their own creation, - they take advantage of the efficiency of this method of communication. That results in a somewhat more efficient use of battery (and possibly data) - when you compare it vs. apps that use IMAP IDLE push.
But, on another hand, I've had a few occasions when in Gmail app, the notification about the new message came about 1 day later, after the message arrived. The scheduled notification in Aquamail (even without Push enabled) arrives on average with a 5 or 7.5 minutes delay (I have different sync setting for different accounts).
PS.
BTW, while writing this post, I just quickly googled (to verify and to provide with the links that explain things in more detail), and the result could be seen at the top of the first page.
Here is "from the horse's mouth":
https://developers.google.com/gmail/api/guides/push- the API for PUSH for Gmail servers (which is implemented in Gmail app).
And there are some useful bits of information in this reddit thread:
https://www.reddit.com/r/apple/comments/37tr55/the_official_gmail_api_finally_gets_push/
