In $$\mathbb {F}_q$$
 , Dartyge and Sarkozy introduced the notion of digits and studied some properties of the sum of digits function. We will provide sharp estimates for the number of elements of special sequences of $$\mathbb {F}_q$$
 whose sum of digits is prescribed. Such special sequences of particular interest include the set of n-th powers for each $$n\ge 1$$
 and the set of elements of order d in $$\mathbb {F}_q^*$$
 for each divisor d of $$q-1$$
 . We provide an optimal estimate for the number of squares whose sum of digits is prescribed. Our methods combine A. Weil bounds with character sums, Gaussian sums and exponential sums.

On the sum of digits of special sequences in finite fields

Arithmetic constraints of polynomial maps through discrete logarithms

The notion of digits in finite fields was introduced a few years ago as an attempt to deliver insight in related yet unresolved questions over the natural numbers. Several such intractable question...

Trace of Products in Finite Fields from a Combinatorial Point of View

Let $q=p^r$ be the power of a prime $p$ and $(\beta_1,\ldots ,\beta_r)$ be an ordered basis of $\mathbb{F}_q$ over $\mathbb{F}_p$. For $$ \xi=\sum\limits_{j=1}^r x_j\beta_j\in \mathbb{F}_q \quad \mbox{with digits }x_j\in\mathbb{F}_p, $$ we define the Rudin-Shapiro function $R$ on $\mathbb{F}_q$ by $$ R(\xi)=\sum\limits_{i=1}^{r-1} x_ix_{i+1}, \quad \xi\in \mathbb{F}_q. $$ For a non-constant polynomial $f(X)\in \mathbb{F}_q[X]$ and $c\in \mathbb{F}_p$ we study the number of solutions $\xi\in \mathbb{F}_q$ of $R(f(\xi))=c$. If the degree $d$ of $f(X)$ is fixed, $r\ge 6$ and $p\rightarrow \infty$, the number of solutions is asymptotically $p^{r-1}$ for any $c$. The proof is based on the Hooley-Katz Theorem.

On the distribution of the Rudin-Shapiro function for finite fields

Many automatic sequences, such as the Thue-Morse sequence or the Rudin-Shapiro sequence, have some desirable features of pseudorandomness such as a large linear complexity and a small well-distribution measure. However, they also have some disastrous properties in view of certain applications. For example, the majority of possible binary patterns never appears in automatic sequences and their correlation measure of order 2 is extremely large. 
Certain subsequences, such as automatic sequences along squares, may keep the good properties of the original sequence but avoid the bad ones. 
In this survey we investigate properties of pseudorandomness and non-randomness of automatic sequences and their subsequences and present results on their behaviour under several measures of pseudorandomness including linear complexity, correlation measure of order $k$, expansion complexity and normality. We also mention some analogs for finite fields.

Pseudorandom sequences derived from automatic sequences.

Prescribing digits in finite fields

Prime numbers with a positive proportion of preassigned digits

Trace of products in finite fields

We study additive double character sums over two subsets of a finite field. We show that if there is a suitable rational self-map of small degree of a set $D$, then this set contains a large subset $U$ for which the standard bound on the absolute value of the character sum over $U$ and any subset $C$ (which satisfies some restrictions on its size $|C|$) can be improved. Examples of such suitable self-maps are inversion and squaring. Then we apply this new bound to trace products and sum-product equations and improve results of the first author and of Gyarmati and S\'ark\"ozy.

Cathy Swaenepoel

Papers

Prescribing digits in finite fields

On the sum of digits of special sequences in finite fields

Prime numbers with a positive proportion of preassigned digits

Trace of products in finite fields

Additive double character sums over some structured sets and applications