Thanks! :)

<?php
include_once 'compat.php';
$user = 0;
$pass = 0;
$mail = 0;
if ($_SERVER['REQUEST_METHOD'] == 'POST' && !empty($_POST['mail']) && !empty($_POST['user']) && !empty($_POST['pass'])) {
    function an($subject){
        if (ctype_alnum($subject)) {
            return true;
        }
    }
    if(an($_POST['user']) && strlen($_POST['user']) <= 20) {
        $user = htmlspecialchars($_POST['user']);
    } else{
        echo 'username is invalid (a-z A-Z 0-9 and less than 20 characters, please)<br>';
    }
    if(filter_var($_POST['mail'], FILTER_VALIDATE_EMAIL ) && strlen($_POST['mail']) <= 255) {
        $mail = htmlspecialchars($_POST['mail')];
    } else {
        echo 'email is invalid<br>';
    }
    if(strlen($_POST['pass']) <= 255) {
        $pass = htmlspecialchars($_POST['pass']);
    } else {
        echo 'password must be less than 255 characters<br>';
    }
    if($pass && $user && $mail) {
        $hash = md5(rand(0,1000));
        require_once('db.php');
        $sth = $db->prepare("
                INSERT INTO user_info (username, password, email, hash)
                VALUES (:username, :password, :mail, :hash)
        ");
        $sth->bindValue(":username", $user, PDO::PARAM_STR);
        $sth->bindValue(":password", password_hash($pass, PASSWORD_DEFAULT), PDO::PARAM_STR);
        $sth->bindValue(":mail", $mail, PDO::PARAM_STR);
        $sth->bindValue(":hash", $hash, PDO::PARAM_STR);
        $success = $sth->execute();
        echo 'You have been successfully signed up :)! To activate your account, click the link in your e-mail account (check your spam box if you don\'t get the e-mail!)<br>
        The last step to activating your account is clicking this link!:<br>
        http://localhost/verify.php?mail='.$mail.'&hash='.$hash.'<br>
        ';
    }
}
?>
<!doctype html>
<html lang="en">
<head>
    <meta charset="utf-8">
    <!--[if lt IE 9]>
    <script src="http://html5shiv.googlecode.com/svn/trunk/html5.js"></script>
    <![endif]-->
</head>

<body>
    <form action="" method="post">
        username: <input type="text" name="user"><br>
        password: <input type="password" name="pass"><br>
        e-mail: <input type="text" name="mail"><br>
        <input type="submit">
    </form>
    <a href="index.php">already have an account?</a>
</body>
</html>