Suppose $X$ is a subset of $\{1, \cdots, n\}$ such that the equation $ax_i+bx_j=cx_k+dx_{\ell}$ where $a+b=c+d,$ $1 \leq a,b,c,d \leq O(n^{1/4 })$ 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 $|X|\geq n^{1-\epsilon},$ where $\epsilon$ is a positive constant depends on $n?$

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