I think there is nothing wrong in appreciating a lecture/ talk by a speaker — one may feel that as their duty/work, but students who enjoy the lecture/talk and learn more from teacher/speaker feel that the speaker has put in some extra effort to present the material in a easily understandable manner. It is a different case with a speaker who is doing it for his own benefit, e.g., an invited visitor or a job talk or a conference talk etc. I personally observed that most non-tenured faculty these days don’t put a lot of effort in teaching and are more interested in research, writing papers, getting grant money etc. to get tenure. In spite of tenure pressure, if a junior faculty is putting a lot of effort in his/her teaching, as a student I cannot stop appreciating.

There is a different between a normal student taking a professor’s course and his/her own Ph.D student — there is a little more to the relationship than just student-professor hierarchy. The additional dimension of culture and country also comes into picture, e.g, in our country a teacher is considered equivalent to God. If my advisor comes to his office to meet me on a Christmas day or New year eve, because I’m stuck with a problem, then he/she is doing more than their duty. In this case, it is important to send a thank you note complimenting their effort. I feel most Ph.D students know the boundary line and even if they do cross that occasionally, it may not be intentional, e.g., to get a favor.

(I think that) it goes with presents as for tips at the restaurant (in France, or as tips over 15% in Canada): I give them (and accept them) when I feel that some work *beyond duty* has been made, and appreciated. It is not so much about a hierarchical relation as about a relation when one perform a job for the other (client or not).

For instance, as the last term ended, I offered a box of chocolates to the 4 TAs of my algorithmic course as they were especially helpful (replacing me on short notice when I got hospitalized, among other things), beyond the description of their job. In the other direction, I gladly accepted a thank you email from a former student and some chocolates from his home country from a current student: those are means of communication, not “bribes”.

Of course, I agree to disagree about what present actually means in each one’s culture: my comment is only on what I would like it to mean

I can’t ever recall being offered a gift from an American student, though several have given a thank you card.

When I read your post, I gathered that you have a much better relationship with your PhD student than with the (stalker?) puzzle giver. In any context, it seems inappropriate to try to re-give a gift that was refused. In fact, it seems creepy. In general, I would refuse gifts if I felt the giver is using the gift to get something or to send a message that I do not want to receive–it is something like refusing to let someone pay on a date if you know you are definitely not interested.

Also, I don’t think the excuse about being in a position of authority is a good one. It sounds quite arrogant, actually. And in this case, there are many other things that are perhaps inappropriate: a student sending you an email complimenting you on a lecture. That would make most professors feel much better than getting a thank you note. But now you might favor the student more.

This is not to say that there are not some gifts, some email messages, some comments, that are inappropriate from a student to a professor. Whatever signals sincere appreciation (or constructive criticism) is okay. Whatever goes beyond this is not okay.