We extend an argument of Felix Behrend to show that fairly dense subsets of the integers exist which contain no solution to certain systems of linear equations.

Using combinatorial methods, we derive several formulas for the volume of convex bodies obtained by intersecting a unit hypercube with a halfspace, or with a hyperplane of codimension 1, or with a flat defined by two parallel hyperplanes. We also describe some of the history of these problems, dating to Polya’s Ph.D. thesis, and we discuss several applications of these formulas.

Let \( \mathbb{F}_2^n \) be the finite field of cardinality \( 2^n \). For all large \( n \), any subset \( A \subset \mathbb{F}_2^n \times \mathbb{F}_2^n \) of cardinality
\[
|A| \gtrsim \frac{4^n \log \log n}{\log n},
\]
must contain three points \( \{(x, y), (x + d, y), (x, y + d)\} \) for \( x, y, d \in \mathbb{F}_2^n \) and \( d \neq 0 \). Our argument is an elaboration of an argument of Shkredov [14], building upon the finite field analog of Ben Green [10]. The interest in our result is in the exponent on \( \log n \), which is larger than has been obtained previously.

We present analytical properties of a sequence of integers related to the evaluation of a rational integral. We also discuss an algorithm for the evaluation of the 2-adic valuation of these integers that has a combinatorial interpretation.

It is proposed that finding the recursion relation and generating function for the (colored) Motzkin numbers of higher rank introduced recently is an interesting problem.

Let \( \mathbb{F}_2^n \) be the finite field of cardinality \( 2^n \). For all large \( n \), any subset \( A \subset \mathbb{F}_2^n \times \mathbb{F}_2^n \) of cardinality \[|A| \gtrsim \frac{4^n \log \log n}{\log n}, \] must contain three points \( \{(x, y), (x + d, y), (x, y + d)\} \) for \( x, y, d \in \mathbb{F}_2^n \) and \( d \neq 0 \). Our argument is an elaboration of an argument of Shkredov [14], building upon the finite field analog of Ben Green [10]. The interest in our result is in the exponent on \( \log n \), which is larger than has been obtained previously.