Slot addressed fans in his match programme notes ahead of Sunday's preseason friendly against Sevilla.
He wrote: "Our tour of the United States was both successful and enjoyable, with a lot of really good work put in on the training pitch and also a chance for those new to the club, such as myself, to get a real taste of the passion and commitment of our global supporters.
“To see the fans in Pittsburgh, Philadelphia and Columbia – at the hotel, at the training sessions and at the matches – was a reminder of this club’s incredible worldwide appeal. I knew it before, of course, but it never hurts to be reminded, and it was certainly a special experience for me as a new head coach.
“Of course, people will naturally look towards the results we had in our friendly matches, and it is true that we enjoyed some good wins against some strong opponents, from which we can take a lot of positives for sure.
“I saw intensity, I saw sharpness and I saw an understanding of the way we want to play – and, of course, to beat both Arsenal and Manchester United, two of our rivals in the Premier League, will never be a bad thing either.
“But I also have high standards, and while I saw lots of things I liked from the team in the USA, I am aware too that there are many areas in which we can, and must, improve and develop. The players know that too – they also have high standards – and I expect to see that improvement in the coming days and weeks, hopefully starting (on Sunday).
“I am, of course, hugely optimistic heading into the new campaign. I said when I arrived at the club that Jürgen left the team in a very good place, and it has been great to be finally able to work with a full squad this week, and to see the quality that we have available.
“Now it is up to us to keep performing like Liverpool did for many years and, hopefully, to bring success to the club. I know that you all want that more than anything, and you can rest assured that myself, my players and my staff will be doing all that we can to deliver for you.
“Let’s get to work...”