# Sidon Sets and Diophantine Equation

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.

• are $a,b,c,d$ fixed? Else there are solutions like $a=c,b=d$, $x_i=x_k$, $x_j=x_l$. – Fedor Petrov Oct 7 '18 at 19:53
• also maybe you need the reverse inequality for $|X|$? – Fedor Petrov Oct 7 '18 at 20:01
• Then "is it true" must be read as "is it possible"? And what is $\epsilon$? – Fedor Petrov Oct 7 '18 at 20:07
• what does it mean "$\epsilon$ is a positive constant depends on $n$?" – Fedor Petrov Oct 7 '18 at 20:53
• 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? – Fedor Petrov Oct 7 '18 at 20:54

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.