Q:

Science fiction story: asymptotic drive the empire lost out to low tech rebels due to focusing on new and ultimately failing techs

In this story the low tech rebels on the outer worlds gradually win small peripheral battles, but at the centre the empire is initially unconcerned as it is about to roll out bigger and better weapons, but each of these has a small but fatal flaw and ultimately they are defeated by the low tech rebels.
It is at least 50 years old.

A:

This is almost certainly Arthur C. Clarke's "Superiority" (1951). It has been asked about on this site and successfully answered several times, for example at Process management and high technology weapon. Novel name and author?
The narrator of the story is highly-placed within the military command structure of the losing side. He describes several inventions that cause the high command to halt operations while they prepare to roll out a new superweapon. The first is a "Sphere of Annhiliation":

Norden proved his case less than a month later, when he demonstrated
  the Sphere of Annihilation, which produced complete disintegration of
  matter over a radius of several hundred meters. We were intoxicated by
  the power of the new weapon, and were quite prepared to overlook one
  fundamental defect - the fact that it was a sphere and hence destroyed
  its rather complicated generating equipment at the instant of
  formation. This meant, of course, that it could not be used on
  warships but only on guided missiles, and a great program was started
  to convert all homing torpedoes to carry the new weapon. For the time
  being all further offensives were suspended.

After this comes the "Battle Analyzer":

We now know that Research had been working on the Battle Analyzer for
  many years, but at the time it came as a revelation to us and perhaps
  we were too easily swept off our feet. Norden's argument, also, was
  seductively convincing. What did it matter, he said, if the enemy had
  twice as many ships as we - if the efficiency of ours could be doubled
  or even trebled? For decades the limiting factor in warfare had been
  not mechanical but biological - it had become more and more difficult
  for any single mind, or group of minds, to cope with the rapidly
  changing complexities of battle in three-dimensional space. Norden's
  mathematicians had analyzed some of the classic engagements of the
  past, and had shown that even when we had been victorious we had often
  operated our units at much less than half of their theoretical
  efficiency.

The third and final innovation is the "Exponential Field":

The final weapon was something so fantastic that even now it seems
  difficult to believe that it ever existed. Its innocent, noncommittal
  name - The Exponential Field - gave no hint of its real
  potentialities. Some of Norden's mathematicians had discovered it
  during a piece of entirely theoretical research into the properties of
  space, and to everyone's great surprise their results were found to be
  physically realizable.
It seems very difficult to explain the operation of the Field to the
  layman. According to the technical description, it "produces an
  exponential condition of space, so that a finite distance in normal,
  linear space may become infinite in pseudo-space." Norden gave an
  analogy which some of us found useful. It was as if one took a flat
  disk of rubber - representing a region of normal space - and then
  pulled its center out to infinity. The circumference of the disk would
  be unaltered - but its "diameter" would be infinite. That was the sort
  of thing the generator of the Field did to the space around it.
As an example, suppose that a ship carrying the generator was
  surrounded by a ring of hostile machines. If it switched on the Field,
  each of the enemy ships would think that it - and the ships on the far
  side of the circle - had suddenly receded into nothingness. Yet the
  circumference of the circle would be the same as before: only the
  journey to the center would be of infinite duration, for as one
  proceeded, distances would appear to become greater and greater as the
  "scale" of space altered.

In each case, the rush to deploy the new technology to the battlefield results in overlooked side effects and negative consequences. This, coupled with the fact that offensive operations are halted to prepare for each deployment, results in the enemy gaining the initiative and overwhelming them with "inferior" but far more plentiful ships and weapons.
The story can be read in its entirety online at Mayo Family.
The "asymptotic drive" that you mention in the question title is another fictional invention of Clarke's, from his novel Imperial Earth. It does not appear in this story, however. Possibly you read the novel and short story around the same time and your memory is conflating the two?

