Additive congruences with factorials modulo a prime

TL;DR

This paper improves bounds on representing any residue modulo a large prime as sums of factorial products, advancing previous exponents, and also provides lower bounds on the size of factorial product sets using additive combinatorics techniques.

Contribution

It introduces improved bounds for representing residues as sums of factorial products and establishes new lower bounds on factorial product set sizes modulo a prime.

Findings

Representation of any residue as sum of two factorial products with exponent 1300/1301.

Representation with five factorial products with exponent 97/113.

Lower bounds on the size of factorial product sets for N<p^{3/5}.

Abstract

Let $p$ be a large prime number. We prove that any integer $\lambda$ modulo $p$ can be represented in the form $$ m!n! +\sum_{i=1}^{47}n_i!\equiv \lambda \pmod p, $$ with $\max\{m,n,n_1,\ldots,n_{47}\}\ll p^{1300/1301}.$ This improves the exponent $1350/1351$ of Garaev, Luca and Shparlinski (2005). Furthermore, we prove that any integer $\lambda$ can be represented in the form $$ m_1!n_1! +m_2!n_2!+m_3!n_3! +m_4!n_4!+m_5!n_5! \equiv \lambda \pmod p $$ with $\max\{m_1,n_1,\ldots,m_5,n_5\}\le p^{97/113 +o(1)}.$ This improves the exponent $27/28$ of Garaev and Garcia (2007). The proofs of these two results are based on the recent work of Grebennikov, Sagdeev, Semchankau and Vasilevskii (2024). We also obtain some lower bound estimates on the cardinality of the product set of two factorials modulo a prime. For instance, we prove that if $N<p^{3/5},$ then $$ \#\{m!n!\pmod p; \, 1\le m,n\le N\}\gg N^{1-o(1)}. $$ The proof of this result is based on works of Banks and Shparlinski (2020), Cilleruelo and Garaev (2016), and the work of Katz and Shen (2008) related to the Ruzsa-Pl\"unnecke inequality from additive combinatorics.