On the distribution of modular inverses from short intervals

For a prime number $p$ and integer $x$ with $\gcd (x,p)=1$ let $\bar{x}$ denote the multiplicative inverse of $x$ modulo $p\mathrm{.}$ In the present paper we are interested in the problem of distribution modulo $p$ of the sequence $$\bar{x},x=1,\dots ,N,$$ and in lower bound estimates for the corresponding exponential sums. As representative examples, we state the following two consequences of the main results. For any fixed $A>1$ and for any sufficiently large integer $N$ there exists a prime number $p$ with $$(\log p{)}^A\asymp N$$ such that $$\underset{\gcd (a,p)=1}{\max }\mid \sum _{x\le N}{\mathbf{e}}_p(a\bar{x})\mid \gg N\mathrm{.}$$ For any fixed positive $\gamma <1$ there exists a positive constant $c$ such that the following holds: for any sufficiently large integer $N$ there is a prime number $p>N$ such that $$N>\exp (c(\log p\log \log p{)}^{\gamma \mathrm{/}(1+\gamma )})$$ and $$\underset{\gcd (a,p)=1}{\max }\mid \sum _{x\le N}{\mathbf{e}}_p(a\bar{x})\mid \gg N^{1-\gamma }\mathrm{.}$$