Comments on a very occasional diary: The Hunt for Addi(c)tive Monster

Nice, yes, that seems to do it. And I made a silly...

2012-01-13T15:47:11.712Z

Nice, yes, that seems to do it. And I made a silly mistake above, when I suggested linearity on yQ contradicted periodicity - of course adding a multiple of s to y will typically carry you off yQ. There will be plenty of periodic solutions, since we can always specify the value 0 on some basis element.

Cheers,
Chris

Dear Chris, thank you for your interest in the po...

2012-01-11T21:52:11.914Z

Dear Chris,

thank you for your interest in the post---it's a pleasure that it helped you. You are absolutely correct that the proof of Statement 1 contains a gap and Engel's device of building a periodic additive function is very ingenious.

However, I think that the proof can be very easily amended: indeed, if an additive function is continuous at 0, it is continuous at arbitrary x, because for any y, such that |x - y| < \delta, |f(x) - f(y)| = |f(x - y)| < \epsilon, where \epsilon and \delta are the same as for continuity at 0.

Dear Nikita, thanks for your post - I enjoyed rea...

2012-01-11T12:09:16.019Z

Dear Nikita,

thanks for your post - I enjoyed reading it, and together with some other sources, it helped me prepare a talk on this subject.

However, I believe there's a gap in your proof of Statement 2. Your proof of Statement 1 only works for x nonzero, because you divide by x/y at the last step. So it's not enough to just prove continuity at zero in order to deduce Statement 2.

There's a nice argument in Engel's "Problem Solving Strategies" (page 273) that begins by using boundedness on (p,p+s) to prove boundedness on (0,s). You then look at g(x)=f(x)-f(s)x/s. This is additive and satisfies g(s)=0, and together these imply g is periodic with period s. Since it's also bounded on (0,s), this means it's bounded on the entire real line, and finally this implies that g is identically zero, which is what we want: if g(y) is nonzero then linearity on yQ leads to a contradiction with boundedness (or periodicity too, I suppose, which is perhaps a bit quicker).

All the best,

Chris