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.

Let \( S \) be a finite set of positive integers with largest element \( m \). Let us randomly select a composition \( a \) of the integer \( n \) with parts in \( S \), and let \( m(a) \) be the multiplicity of \( m \) as a part of \( a \). Let \( 0 \leq r < q \) be integers, with \( q \geq 2 \), and let \( p_{n,r} \) be the probability that \( m(a) \) is congruent to \( r \mod q \). We show that if \( S \) satisfies a certain simple condition, then \( \lim_{n \to \infty} p_{n,r} = 1/q \). In fact, we show that an obvious necessary condition on \( S \) turns out to be sufficient.

This is an attempt of a comprehensive treatment of the results concerning estimates of the \( L^1 \)-norms of linear means of multiple Fourier series, the Lebesgue constants. Most of them are obtained by estimating the Fourier transform of a function generating such a method. Frequently the properties of the support of this function affect distinctive features in behavior of these norms. By this geometry enters and works hand-in-hand with analysis; moreover, the results are classified mostly in accordance with their geometrical nature. Not rarely Number Theory tools are brought in. We deal only with the trigonometric case – no generalizations for other orthogonal systems are discussed nor are applications to approximation. Several open problems are posed.

Let \( G \) be a finite abelian group and \( E \) a subset of it. Suppose that we know for all subsets \( T \) of \( G \) of size up to \( k \) for how many \( x \in G \) the translate \( x + T \) is contained in \( E \). This information is collectively called the \( k \)-deck of \( E \). One can naturally extend the domain of definition of the \( k \)-deck to include functions on \( G \). Given the group \( G \), when is the \( k \)-deck of a set in \( G \) sufficient to determine the set up to translation? The \( 2 \)-deck is not sufficient (even when we allow for reflection of the set, which does not change the \( 2 \)-deck) and the first interesting case is \( k = 3 \). We further restrict \( G \) to be cyclic and determine the values of \( n \) for which the \( 3 \)-deck of a subset of \( \mathbb{Z}_n \) is sufficient to determine the set up to translation. This completes the work begun by Grünbaum and Moore [GM] as far as the \( 3 \)-deck is concerned. We additionally estimate from above the probability that for a random subset of \( \mathbb{Z}_n \), there exists another subset, not a translate of the first, with the same \( 3 \)-deck. We give an exponentially small upper bound when the previously known one was \( O(1/\sqrt{n}) \).

Bourgain’s theorem says that under certain conditions a function \( f : \{0,1\}^n \to \{0,1\} \) can be approximated by a function \( g \) which depends only on a small number of variables. By following his proof we obtain a generalization for the case that there is a nonuniform product measure on the domain of \( f \).

Given integers \( s, t \), define a function \( \phi_{s,t} \) on the space of all formal series expansions by \(\phi_{s,t}\left(\sum a_n x^n\right) = \sum a_{sn+t} x^n.\) For each function \( \phi_{s,t} \), we determine the collection of all rational functions whose Taylor expansions at zero are fixed by \( \phi_{s,t} \). This collection can be described as a subspace of rational functions whose basis elements correspond to certain \( s \)-cyclotomic cosets associated with the pair \( (s, t) \).

In this note we use the theory of theta functions to discover formulas for the number of representations of N as a sum of three squares and for the number of representations of N as a sum of three triangular numbers. We discover various new relations between these functions and short, motivated proofs of well known formulas of related combinatorial and number-theoretic interest.