Pink Letter was written and sent by Mance

[NotSoSilentSister]

Also, I really think people tend to overestimate "Arya"'s strategical importance.

That sham obviously never really worked anyway - at the end of Dance, lots of the Bolton's Northern allies are already plotting to betray them (Manderleys, Umbers, maybe even Dustins, etc.) and that's before "Arya" got kidnapped. Now, maybe the Manderleys can hide their betrayal in the chaos and confusion of that battle for a while longer to support Stannis faking his death, etc. but the Umber situation at least has likely blown up already at the point of letter writing. So Ramsay should be well aware that his Northern allies never bought into the whole charade anyway. If at this point he still thinks that getting back "Arya" would fix this problem, he lacks even the low cunning his worst critics still seem to credit him with.

[Muggle]

Your false king is dead, bastard. He and all his host were smashed in seven days of battle. I have his magic sword. Tell his red whore. Your false king’s friends are dead. Their heads upon the walls of Winterfell. Come see them, bastard. Your false king lied, and so did you. You told the world you burned the King-Beyond-the-Wall. Instead you sent him to Winterfell to steal my bride from me. I will have my bride back. If you want Mance Rayder back, come and get him. I have him in a cage for all the north to see, proof of your lies. The cage is cold, but I have made him a warm cloak from the skins of the six whores who came with him to Winterfell. I want my bride back. I want the false king’s queen. I want his daughter and his red witch. I want his wildling princess. I want his little prince, the wildling babe. And I want my Reek. Send them to me, bastard, and I will not trouble you or your black crows. Keep them from me, and I will cut out your bastard’s heart and eat it.

It was signed,

Ramsay Bolton,

Trueborn Lord of Winterfell.

Wildlings call members of the watch crows just saying....

Its wishful thinking but i hope Mance is still alive. I enjoy him

[The Mother of The Others]

Maybe it's a pre-arranged code from Mance to Jon, saying you better get here within "seven days," and letting Jon know there was no Arya hostage situation anymore so Jon is free to attack. If penned by Mance, he's passing it off as something Ramsay would have sent--this keeps Melisandre's secret (that Mance is dead). And while disguised as Ramsay, Mance basically gave Jon a forecast of how things were about to be unless Jon did something to change it.