On Sums of Units

On the length of arithmetic progressions in linear combinations of S-units

“…Our main result is an explicit upper bound for L depending only on r, t and n. Our argument is different from the ones from [9] and [10] and avoids the use of van der Waerden's theorem, and so it is clearly much smaller than the ones which would follow from the works [9] and [10]. We note that a relatively small, and completely explicit upper bound for L is important also for the applications.…”

“…For the theory and applications of such and related equations we refer to [3,4,5,6,7,8], and the references therein. Recently, Hajdu [9], and Jarden and Narkiewicz [10], independently, have investigated arithmetic progressions in the linear combinations of elements from such groups Γ. Their results had found several applications to Diophantine problems.…”

“…. , 1)}) has been obtained by Jarden and Narkiewicz [10]. Although recent, these results have already found many applications to problems coming from different parts of number theory: to the so-called unit sum number problem (see [10]), to a question of M. Pohst about representing primes as sums or differences of powers of 2 and 3 (see [9]), and to bound the lengths of arithmetic progressions in the solution sets of norm form equations (c.f.…”

“…Interestingly, aside from a result of Evertse, Schlickewei and Schmidt [6] on the number of nondegenerate solutions to linear equations with unknowns from a finitely generated group, the proofs of the theorems of Hajdu [9] and Jarden and Narkiewicz [10] are also based upon a classical result of van der Waerden [12] concerning monochromatic arithmetic progressions. Another common feature of the results of [9] and [10] is that the upper bounds for L are not explicitly given.…”

“…Another common feature of the results of [9] and [10] is that the upper bounds for L are not explicitly given. One could go through the above papers and write down an upper bound for L based on their arguments, but since the proofs use van der Waerden's result, it is quite likely that the upper bound one would end up in this way with will be huge.…”