NOTE: Leaving this up for reference, but Jeremy Crawford has indicated they do intend to publish the change that:
Arcane Recovery—"once per day" should say you have to finish a long rest between uses.
As of the November 2018 Players Handbook errata, however, this change has not been incorporated into the rules as written.
Arcane recovery does not require a long rest between uses
Recent errata have clarified when "once a day" actually means "once between long rests" for some powers - but notably not for Arcane Recovery. These edits were not done ad hoc, but in preparation for a revised printing of the Players Handbook. The errata state "Recent printings of the book include revised text that reflects the explanations here."
As Arcane Recovery still reads "once a day," I would argue the most reasonable conclusion is that this was an intentional non-edit. (NOTE: This conclusion was wrong.) So, no long rest is required between uses of Arcane Recovery.
It would be a little strange for a (long) rest to be required to gain the benefits of a (short) rest.
"Once a day" and "between long rests" are not synonyms
In the revised Player's Handbook, powers that recharge between long rests, such as Drow Magic, no longer mention "once a day." That language has been revised to read "regain the ability when you finish a long rest." (PH, Dark Elf, p. 24). The rules for recharging these powers were really changed, not clarified. There's no justification for conflating the terms "day" and "between long rests" (although a long rest may of course last until the next day has begun).
Every 24 hours isn't it either
If we take the rules at face-value, there is also nothing in the rules that says the uses of Arcane Recovery need to be greater than or equal to 24 hours apart. A wizard might use arcane recovery on Monday evening, and then on Tuesday morning, without exceeding "once a day."
So, what exactly is "a day" then?
The only tricky part remaining is to determine when one day ends and another begins. Arcane Recovery is silent on this matter, but many magic items in the DM Guide recover charges "at dawn" (see Staff of Power, DM Guide, p. 202).
(Beyond the hint we get from the DM Guide, the term "day" is undefined in the rules. In cases where the rules do not define a term, the common dictionary definition will have to suffice.)
If the availability of Arcane Recovery resets at dawn as well, that contributes to the feel of wizards and the magic items deriving their power from the same "weave of magic" (PH p. 205).
But you might choose a different time, or have different wizards regain their power at different times. For example, midnight (a.k.a., the witching hour) might be an appropriate time.