Given two integers integers $0<b<p$ and a real $\alpha\in(0,1)$ what is the largest $m$ we have such that in the interval $(p^\alpha,p-p^\alpha)$ there are $m$ integers $a\_1<\dots<a\_m$ such that


1. $a\_i-a\_j\neq a\_{i'}-a\_{j'}$ holds if $i\neq i'$ or $j\neq j'$ or both.
2. $\min\_{1\leq i<j\leq m}|a\_i-a\_j|>b$
holds?



An extremal combinatorics problem
Taking [Fedor Petrov](https://mathoverflow.net/users/4312/fedor-petrov)'s observation a little further, I believe the right question to ask is as follows: 
 $$ \text{What is the largest size of a $b$-separated Sidon set in the interval $[0,L]$?} $$ 
Here *"$b$-separated"* means that the difference between any two consecutive elements of the set is at least $b+1$.


Clearly, one cannot find in $[0,L]$ a $b$-separated set of size larger than $L/b+1$; on the other hand, taking a large Sidon set in $[0,L/(b+1)]$ and dilating it by the factor $b+1$, you can get a $b$-separated Sidon set in $[0,L]$ of size $(1+o(1))\sqrt{L/(b+1)}$.


In general, the answer will depend heavily on the relation between $L$ and $b$. If $b$ is small as compared to $L$ (say, $b<\sqrt{L/2}$), then you can find a $b$-separated Sidon set $S\subset[0,L]$ of size about $\sqrt{L/2}$, as follows. 


Let $q$ be a prime satisfying $2q^2\le L$ (this is the size of our set $S$, so we want to choose $q$ as large as possible subject to this condition). Fix a primitive root $g$ modulo $q$, and set
 $$ S := \{2qn+\nu(n)\colon 1\le n\le q-1 \}, $$
where $\nu(n)$ is the integer in the range $[0,q-2]$ such that $g^{\nu(n)}\equiv n\pmod q$. 


Since $2q(n+1)>(2qn+q-2)+b$ (as it follows from $q\approx\sqrt{L/2}>b$), the set $S$ is $b$-separated.


To see that $S$ is Sidon, notice that 
 $$ (2qn\_1+\nu(n\_1))+(2qn\_2+\nu(n\_2)) = (2qn\_3+\nu(n\_3)) + (2qn\_4+\nu(n\_4)) $$ 
yields $n\_1+n\_2=n\_3+n\_4$, leading to $\nu(n\_1)+\nu(n\_2)=\nu(n\_3)+\nu(n\_4)$ and, consequently, to $n\_1n\_2\equiv n\_3n\_4\pmod q$. Thus, 
 $$ n\_1+n\_2 =n\_3+n\_4 $$
and
 $$ n\_1n\_2 \equiv n\_3n\_4\pmod q, $$
as a result of which $\{n\_1,n\_2\}=\{n\_3,n\_4\}$. 


