3
$\begingroup$

Suppose $X$ is a subset of $\{1, \cdots, n\}$ such that the equation $ax_i+bx_j=cx_k+dx_{\ell}$ where $a+b=c+d,$ $a,b,c,d \in \mathbb{N}$ and $x_i, x_j, x_k, x_{\ell} \in X,$ has only trivial solution. A solution is trivial if $x_i=x_j=x_{k}=x_{\ell}.$

What can we say about the size of $X?$ Is this possible that $|X|\geq n^{1-o(1)}$?

I think the answer is related to Sidon Sets, but I could not find any references. Any help is greatly appreciated.

$\endgroup$
7
  • $\begingroup$ are $a,b,c,d$ fixed? Else there are solutions like $a=c,b=d$, $x_i=x_k$, $x_j=x_l$. $\endgroup$ Oct 7, 2018 at 19:53
  • $\begingroup$ also maybe you need the reverse inequality for $|X|$? $\endgroup$ Oct 7, 2018 at 20:01
  • $\begingroup$ Then "is it true" must be read as "is it possible"? And what is $\epsilon$? $\endgroup$ Oct 7, 2018 at 20:07
  • $\begingroup$ what does it mean "$\epsilon$ is a positive constant depends on $n$?" $\endgroup$ Oct 7, 2018 at 20:53
  • $\begingroup$ in any case: you may construct the set by adding the elements one by one, this allows to get about $c\cdot n^{1/3}$ elements for free. Is it enough? $\endgroup$ Oct 7, 2018 at 20:54

1 Answer 1

6
$\begingroup$

Such $X$ do indeed exist, and are explicitly constructed in I.Z. Ruzsa, Solving a linear equation in a set of integers, Acta Arith., LXV.3 (1993), pp. 259-282, Theorem 7.5. The whole paper is devoted to upper and lower bounds on sizes of solution-free sets to equations such as the one you are asking about. A mathscinet forward search from that paper should yield further results.

$\endgroup$
3
  • 5
    $\begingroup$ specifically Theorem 7.5 in this paper seems to the answer original question $\endgroup$ Oct 8, 2018 at 14:36
  • $\begingroup$ Thank you, Fedor! I knew that the answer was in there, but did not have time this morning to find the exact place. I have updated the answer. $\endgroup$
    – Alex B.
    Oct 8, 2018 at 16:10
  • $\begingroup$ @Kim: you can just ask a separate question here on MO. I am unlikely to be of much help. It has been 13 years since I read that paper - I was a 2nd year undergraduate. $\endgroup$
    – Alex B.
    Oct 24, 2018 at 21:03

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.