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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 $|X|\geq n^{1-o(1)}$?

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

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.

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.

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 $|X|\geq n^{1-o(1)}$?

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

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 true that $|X|\geq n^{1-\epsilon}$?

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

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.