Gowers norms control Diophantine inequalities

Abstract

Abstract. A central tool in the study of systems of linear equations with integer coefficients is the Generalised von Neumann Theorem of Green and Tao. This theorem reduces the task of counting the weighted solutions of these equations to that of counting the weighted solutions for a particular family of forms, the Gowers norms f U s+1 [N ] of the weight f . In this paper we consider systems of linear inequalities with real coefficients, and show that the number of solutions to such weighted diophantine inequalities may also be bounded by Gowers norms. Further, we provide a necessary and sufficient condition for a system of real linear forms to be governed by Gowers norms in this way. In a forthcoming paper we will discuss the case in which the weights are unbounded but suitably pseudorandom, with applications to counting the number of solutions to diophantine inequalities over the primes.

Bootstrapping partition regularity of linear systems

“…(Other introductions may be found in many places, including [20, § 4], [24, § § 2 and 3], [41, Appendix A], and [29, § 1]. All these, including [14], ultimately refer to [21] for details, though the paper [41] does expand on the details somewhat in § 4. )…”

Linear inequalities in primes

“…We will be concerned with diophantine inequalities, a topic that we first considered in [21]. Before giving our first main result (Theorem 1.7) let us briefly review some previous results concerning diophantine inequalities in the primes.…”

“…Our first result does not concern the most general type of diophantine inequality, but nonetheless it enjoys several applications. To state it, we recall the notion of the dual degeneracy variety, which we defined in Definition 2.3 of [21] in order to manipulate the non-degeneracy conditions more succinctly. Definition 1.5 (Dual degeneracy variety, [21]).…”

“…To state it, we recall the notion of the dual degeneracy variety, which we defined in Definition 2.3 of [21] in order to manipulate the non-degeneracy conditions more succinctly. Definition 1.5 (Dual degeneracy variety, [21]). Let m, d be natural numbers satisfying d m + 2.…”

“…For example, the matrix 1 −2 1 0 2 −4 0 √ 3 is in V * degen (2,4), since the vector (0, 0, −2, √ 3) lies in its row space. As is explained at length in [21], if one wishes to count solutions to an inequality given by L using a method involving Gowers norms, then one can only possibly succeed if L / ∈ V * degen (m, d). Returning to Theorem 1.1, we observe that the nondegeneracy condition in the statement of that theorem is exactly the condition that L / ∈ V * degen (m, d).…”