On finding many solutions to S-unit equations by solving linear equations on average

Abstract

We give improved lower bounds for the number of solutions of some Sunit equations over the integers, by counting the solutions of some associated linear equations as the coefficients in those equations vary over sparse sets. This method is quite conceptually straightforward, although its successful implementation involves, amongst other things, a slightly subtle use of a large sieve inequality. We also present two other results about solving linear equations on average over their coefficients.

Many solutions to the $S$-unit equation $a+1=c$

“…with all their prime factors lying in S. Konyagin and Soundararajan [11] showed that this special case too has exponentially many solutions for certain well chosen sets S. Namely, they showed that there are sets S for which the equation a + 1 = c has at least exp(s 1/16 ) solutions. This was subsequently improved by Harper [7] who showed the existence of sets S for which there are at least exp(s 1/6−ǫ ) solutions. In this paper we make further progress on this question, by showing that there are sets S with at least exp(s 1/4 / log s) solutions.…”