After several days testing and trying out, I am confirming that it now works!

Thanks a lot guys!

Can you try what I just committed?

You will need to run the new migration - this can be done from Admin->Config

You *might* also need to re-initialize the AS account depending on how 
messed up the client's database is.

Changes have been made in Git for this ticket:

Ignore duplicate incoming message additions (Bug: 10644)
If incoming changes are successfully applied, but the client does
not receive (or ignores) the new synckey, the next SYNC command will
send duplicate client changes. We now save the incoming clientid
string in the map table so we can ignore duplicate additions in these cases.

Note the new migration that must be run

  4 files changed, 85 insertions(+), 25 deletions(-)
http://git.horde.org/horde-git/-/commit/7d9499c647b1f12703da34f980a91a23d7fd16fa

Nope. Your logs indicate it is (at least partially an ActiveSync) 
issue. There may be some kind of conflict with the App you mentioned 
that is causing the device to not receive the synckey (though I doubt 
it), but the AS server still needs to be able to handle this 
situation. Working on a solution as we speak...

I *think* that I've found it, and if I'm right, you can close this bug.

You see, I am using a third party app on the iphone called "Week 
Calendar".  I started to think that the problem might be on the iphone 
end and suspected the app to have bugs.

Going on the App store and searching for my app, I've found this link:

http://www.utilitap.com/weekcalendar/faq.php?item=duplicate

It mentions MobileMe, which I am not using, but the bug is the same.   
I will email them to ask for further assistance and see that they say.

From what I can tell, Horde receives all the changes from the device, 
processes them correctly and then sends the new synckey to the device 
(which indicates the changes were successful). For some reason, the 
device is not accepting (or receiving) the response because it 
immediately re-issues the same SYNC request (including the on-device 
changes). No idea why your device is not accepting the request.

Unable to reproduce your symptoms but will try to track down based on 
log and errors.

I am again experiencing problems with AS but this time I know whats 
going on better.  Again, some events are not properly sync'ed and are 
duplicated.

I discovered the following facts:

-Events that were already in kronolith are kept OK.  No problem with them.
-Events that I CREATED on the mobile device while I was NOT online 
with my horde installation (out of range from my wifi LAN) are not 
properly sync'ed.  They are sync at the right moment (day, time) but 
duplicated.
-I am not sure about events that were modified on the mobile device 
and later sync'ed as I did not had a chance to try yet.
-Events that created/modified on the mobile device WHILE connected to 
the local network are sync'ed properly.

In other words, only  (SO FAR) events that are created from scratch 
offline are not properly sync'ed.

Attached is the AS log for today.  I deleted the entries that were 
dated prior to today, because the log was well over 5MB.

While removing my private data from the log, I've noticed 3 strange things:

1-The events that were duplicated by AS were in the log twice.  I 
marked them as:  !!!Faulty Event!!!
2-I did NOT change anything with my contacts since original sync (when 
I created the exchange account on the mobile dev) yet there is 
references to contact changes in the AS log.
3-One line caught my attention .  Not sure if it has anything to do 
with my problem:

2011-10-12T18:01:45-04:00 NOTICE: [Appl6Q109352A4S] Error saving state 
for synckey {xxxxxxxx-xxxx-xxxx-xxxx-xxxxxxxxxxxx}11: SQLSTATE[23000]: 
Integrity constraint violation: 1062 Duplicate entry 
'{xxxxxxxx-xxxx-xxxx-xxxx-xxxxxxxxxxxx}11' for key 'PRIMARY' - 
removing previous sync state and trying again.

Also if important to mention: I did not remove or delete anything 
dated 2011-10-12.
Help is appreciated.
Thanks to the staff!