Permutation binomials over finite fields. Conditions of existence

Let $1\leq j<i\leq 2^n-1$, $1\leq k\leq2^n- 1$, $\alpha$ is a primitive element of the field $\mathbb F_{2^n}$. It is proved that: 1) if a function $f\colon\mathbb F_{2^n}\to\mathbb F_{2^n}$ of the form $f(y)=\alpha^ky^i+y^j$ is one-to-one function, then $\operatorname{gcd}(i-j,2^n-1)$ doesn't divide $\operatorname{gcd}(k,2^n-1)$; 2) if $2^n-1$ is prime, then one-to-one function $f\colon\mathbb F_{2^n}\to\mathbb F_{2^n}$ of the form $f(x)=\alpha^kx^i+x^j$ doesn't exist; 3) if $n$ is a composite number, then there is one-to-one function $f\colon\mathbb F_{2^n}\to\mathbb F_{2^n}$ of the form $f(x)=\alpha^kx^i+x^j$; 4) if $2^n-1$ has a divisor $d<\frac n{2\log_2(n)}-1$, then there is one-to-one function $f\colon\mathbb F_{2^n}\to\mathbb F_{2^n}$ of the form $f(y)=ay^i+y^j$ for some $a\in\mathbb F^*_{2^n}$, $0<j<i<2^n-1$.