We present an infinite family of Borwein type $+ - - $ conjectures. The expressions in the conjecture are related to multiple basic hypergeometric series with Macdonald polynomial argument.
In the context of the (generalized) Delta Conjecture and its compositional form, DAdderio, Iraci, and Wyngaerd recently stated a conjecture relating two symmetric function operators, $D_k$ and $Theta_k$. We prove this Theta Operator Conjecture, findi
ng it as a consequence of the five-term relation of Mellit and Garsia. As a result, we find surprising ways of writing the $D_k$ operators.
We give a new proof of a sumset conjecture of Furstenberg that was first proved by Hochman and Shmerkin in 2012: if $log r / log s$ is irrational and $X$ and $Y$ are $times r$- and $times s$-invariant subsets of $[0,1]$, respectively, then $dim_text{
H} (X+Y) = min ( 1, dim_text{H} X + dim_text{H} Y)$. Our main result yields information on the size of the sumset $lambda X + eta Y$ uniformly across a compact set of parameters at fixed scales. The proof is combinatorial and avoids the machinery of local entropy averages and CP-processes, relying instead on a quantitative, discrete Marstrand projection theorem and a subtree regularity theorem that may be of independent interest.
In this paper we set up a bivariate representation of partial theta functions which not only unifies some famous identities for partial theta functions due to Andrews and Warnaar, et al. but also unveils a new characteristic of such identities. As fu
rther applications, we establish a general form of Warnaars identity and a general $q$--series transformation associated with Bailey pairs via the use of the power series expansion of partial theta functions.
We prove the test function conjecture of Kottwitz and the first named author for local models of Shimura varieties with parahoric level structure, and their analogues in equal characteristic.
The Markoff injectivity conjecture states that $wmapstomu(w)_{12}$ is injective on the set of Christoffel words where $mu:{mathtt{0},mathtt{1}}^*tomathrm{SL}_2(mathbb{Z})$ is a certain homomorphism and $M_{12}$ is the entry above the diagonal of a $2
times2$ matrix $M$. Recently, Leclere and Morier-Genoud (2021) proposed a $q$-analog $mu_q$ of $mu$ such that $mu_{qto1}(w)=mu(w)$ is the Markoff number associated to the Christoffel word $w$. We show that for every $q>0$, the map ${mathtt{0},mathtt{1}}^*tomathbb{Z}[q]$ defined by $wmapstomu_q(w)_{12}$ is injective over the language $mathcal{L}(s)$ of a balanced sequence $sin{mathtt{0},mathtt{1}}^mathbb{Z}$. The proof is based on new equivalent definitions of balanced sequences.