This thesis will consist of three main chapters.

It is a standard fact of linear algebra that every matrix with rank k contains a k ⨯ k submatrix with rank k. In Chapter 2 we generalise this fact asymptotically to a class of notions of rank for higher-order tensors, containing in particular the tensor rank, the slice rank and the partition rank. We show that for every integer d ⩾ 2 and every notion R in this class of notions of rank, there exist functions F and G such that if an order-d tensor has R-rank at least G(l) then we can restrict its entries to a product of sets X$\mathrm{<mrow\; data-mjx-texclass="ORD">1</mrow>}$ ⨯ … ⨯ X such that the restriction has R-rank at least l and the sets X$\mathrm{<mrow\; data-mjx-texclass="ORD">1</mrow>}$,…,X each have size at most F(l). Combining the proof methods that we use to prove this result with a few additional ideas then allows us to show that under a very natural condition we can furthermore require the sets X$\mathrm{<mrow\; data-mjx-texclass="ORD">1</mrow>}$,…,X to be pairwise disjoint.

In Chapter 3 we extend to the case of restricted subsets a result of Green and Tao on the equidistribution of high-rank polynomials over finite prime fields. We show that for every fixed prime integer p, for every integer d ∈ [2, p-1], and for every non-empty subset S of F, it is true uniformly in n that if P: F → F is a polynomial with degree at most d such that P(x) is not approximately equidistributed on F when x is chosen uniformly at random in S, then P coincides on S with a polynomial which can be expressed as a function of a bounded number of polynomials of degree at most d-1. Our argument uses two results which are known by that point: the second main result of Chapter 2, and the fact that an order-d tensor over F with high partition rank necessarily has high analytic rank.

In Chapter 4 we prove approximation results for conditions on {0,1} and similar sets when those conditions are defined using polynomials from F to F for some prime p. We show in particular that for every non-empty subset S of F, if for some linear forms φ on F and some subsets E of F, the set U of all x ∈ S satisfying all conditions φ(x) ∈ E is dense inside S then there exist a bounded number of pairs (θ, T), where the θ are linear forms on F and the T are subsets of F, such that the set of x ∈ S satisfying all conditions θ(x) ∈ T is contained in U and has inside S approximately the same density as U has inside S. As an application, we rule out a class of potential counterexamples to a first unsolved case of the polynomial density Hales-Jewett conjecture. We also generalise our approximation results in some other directions: in particular we deduce an approximation result (with a weaker formulation) for polynomials of small degree from the main result of Chapter 3.