I am not a linguist, but I receive linguistics classes at university. I can try to explain a couple things.
Essentially, here is what H.A did. He analyzed simple sentences such as those:
"Stop singing!" or "I hate singing!".
He noticed that in order for someone to say "stop doing something", the "doing" in question has to have started, in other words, it has to be taken for granted.
Similarly, He said that for someone to like or hate something, they need to have done it in the past and tried it.
He concluded that '-ing' is a marker which indicates that the utterer takes for granted the action or event involved.
Therefore: "I leave tomorrow" announces that the utterer will leave the day before.
"I am leaving tomorrow" on the contrary takes it for granted, the utterer is giving a reminder or shows that he already thought about leaving before, it is not new to him.
H.A uses the terms "rhematic" (new information) and "thematic" (old information).
He moved on to use his theory in other cases:
"He starts to sing" announces what the person starts to do while "he starts singing" involves an already envisaged action, the "singing" per se is not surprising and was to be expected.
"We will land soon" would be used to announce that the landing will happen soon while "We will be landing soon" would simply be a reminder or in any case, would show the event as normal and "given".
"I heard a dog bark" and "I heard a dog" barking would not differ according to whether or not you heard the whole action, like traditional grammar would hold, but instead "I heard a dog bark" would announce that the dog was barking while " I heard a dog barking" gives the barking of the dog an already pre-established dimension thanks to the marker 'ing'.
His theory holds merits and can help in cases where it is hard to choose between a simple or a progressive form.
But it is hard to argue against the fact that there is an aspectual difference between "I wrote a letter" in which the writing is complete and "I was writing a letter" which does not indicate that the writing was complete.
Therefore, his theory is regarded as intellectually enriching but does have some faults.
Please note that I gave an extremely simplified interpretation of this theory so that you can understand the basic idea behind it. If you wish to understand it more in detail, please tell me.