Amazon has contacted some of its US customers to offer partial refunds for e-books bought between April 2010 and May 2012.

The compensation – $0.30-$1.32 per title – is the result of a settlement between publishers and the US authorities.

Barnes & Noble is expected to email its customers with a similar offer soon.

Several e-book publishers – plus Apple – were accused of colluding to fix and raise prices.

Customers covered by the settlement will not receive refunds until a hearing approving its terms takes place in February next year.

The compensation will cover titles produced by three major publishers – Hachette, HarperCollins and Simon & Schuster. The firms have together raised a fund of $69 million to pay the fees.

Two other publishers, Penguin and Macmillan, declined to agree to the settlement and will instead be taken to court, along with Apple.

Compensation will increase depending on the number of titles bought by a customer. While the exact amounts are yet to be confirmed, they will differ depending on whether or not a title appeared in the New York Times best-seller list.

In an email to customers sent over the weekend, Amazon said: “We have good news.

“You are entitled to a credit for some of your past e-book purchases as a result of legal settlements between several major e-book publishers and the attorneys general of most US states and territories, including yours.”

The company added: “In addition to the account credit, the settlements impose limitations on the publishers’ ability to set e-book prices.

“We think these settlements are a big win for customers and look forward to lowering prices on more Kindle books in the future.”

The case, which was brought in April, was fuelled by several major publishers’ decision to change how they worked with e-book retailers.

Physical books are typically sold at a wholesale price, where retailers buy on bulk and are then free to set their own prices.

E-books were also sold in this way until five publishers decided to switch instead to an agency model. Under this system, publishers set the price of a book and the retailer selling it gets a 30% cut.

This method was seen as a way of curbing Amazon’s dominance of e-book sales on its Kindle platform – and was said to be favored by the late Apple founder Steve Jobs.

According to a biography published after his death, Steve Jobs once said: “We were not the first people in the books business.

“Given the situation that existed, what was best for us was to do this aikido move and end up with the agency model. And we pulled it off.”